The theory of the expanding universe is so widely accepted, that the redshift is sometimes used as a measure of distance to far away galaxies.
But is it still possible that the redshift is caused by some unknown phenomena and not by galaxies moving away from each other?
Is there any other proof (apart from the redshift) that the universe is indeed expanding and far away galaxies are moving away from us?
Yes, there is direct, non-red-shift evidence of expansion.
The past temperature of the Cosmic Microwave Background Radiation (CMBR) has been directly measured and found to be substantially higher than it is today. Its reduction in temperature over time is direct evidence of expansion. Here are the details:
According to this paper, the CMBR was measurably hotter in the past (less-technical synopsis here). The researchers observed absorption lines in a gas cloud located in a distant galaxy and found that the pattern of lines seen could only be explained if the CMBR temperature at time of absorption was between 6 K and 14 K (it's 3 K now). This temperature is consistent with the expected temperature for that galaxy's redshift (9 K). Note that the temperature was measured from the specific pattern of lines seen and not from how much the lines had been red-shifted; this measurement would yield the same temperature even if there were no red-shifting. Since a hotter temperature implies a higher density, this cooling of the CMBR over time is direct evidence for expansion of the universe.
Additional comments
What is the relationship between red-shift and absorption lines?
Inspired by a conversation with uhoh in the comments:
In my answer I refer to a "pattern" of "absorption lines". For those not versed in the topic allow me to explain.
When a light shines through a cloud of gas specific frequencies of light get absorbed. When this light is then shone through a prism the blocked frequencies will appear as black lines in the spectrum (see illustration below). The exact lines that appear and their positions in the spectrum (the "pattern" of "absorption lines") depends on the elements present in the gas and the gas's environment. The effect is most clearly seen with a light that emits photons at all frequencies; this kind of light is known as black-body radiation. Although emitting light at all frequencies, a black-body radiator will emit the most light at a particular wavelength; the location of this peak is referred to as the black-body's temperature. The CMBR discussed in the question is an example of black-body radiation.
Source: Doppler Shift, Edward L. Wright
(Excellent site BTW, the FAQ is worth a look for more info on red shifts and cosmology in general)
As light is travels through (expanding) space it's wavelength and the wavelengths of the absorption lines stretches at a fixed rate for all frequencies. Let's say that at time of emission/absorption a spectrum shows lines at wavelengths of 1, 3, and 5 nm1. After the photons have traveled for a certain amount of time, all of the spectrum's wavelengths will appear to have doubled2. The line formerly at 1 nm is now seen at 2 nm, the one formerly at 3 nm is now seen at 6 nm, and the one originally at 5 nm is now seen at 10 nm. Though their absolute frequencies change over time, the ratio of the lines' wavelengths (and frequencies) relative to one-another remains constant.
The precise amount that a given object's spectrum is shifted directly correlates with its distance. As seen in the diagram above, close objects (like the Sun) show no red-shifting. As one looks at objects further and further away one sees increasing amounts of red shift3.
In the discussion in the answer above, it's this pattern of relative positions in the lines that is affected by the CMBR temperature at time of absorption and not the degree to which the lines have been shifted.
1 To put it technically, this point is at $z=0$ where $z$ indicates the magnitude of the shift, positive for red shifts (moving away) and negative for blue shifts (approaching). A more in-depth discussion of this topic (including the precise definition of $z$) can be found here.
2 The wavelength-doubling (frequency-halving) point is at $z=1$
3 It should be noted that since there is some uncertainty in the rate that the Universe is expanding at, red-shifts do not refer to precisely-known distances. Thus astronomers and cosmologists rarely refer to the distances to distant objects in absolute terms of, say, light-years or parsecs, preferring, rather, to use the amount of red-shift observed (the $z$ mentioned above).
The mechanism behind red shifting isn't that the photons themselves are changing rather it is that the very space that the electromagnetic waves are moving through is expanding. (Photons are both particles and waves; no, it's not exactly intuitive.) This constant stretching of space stretches the light's wavelength giving rise both to the effect of red-shifting and the increase in a given photon's red shift over time.
Douglas Hofstadter, CC A-SA 3.0
How does red-shift relate to the CMBR?
In the comments Alchimista asked "Isn't CMBR actually the quintessence of red-shift anyway?"
(I'm assuming you're using the common, and not cosmological, meaning of "quintessence")
Yes, the current CMBR temperature (3 K) is generally agreed to be the result of relatively high-energy photons (3000 K) emitted about 380,000 years after the Big Bang that have had their wavelengths stretched over time by the expansion of the Universe towards the red (i.e. cooler or lower-energy) end of the spectrum. This expansion was inferred by Hubble et al. from the observation that smaller and dimmer galaxies (as seen from Earth) have a greater shift in their spectra. The farther the apparent distance, the greater the observed shift. Using this apparent distance-correlated red shift we can infer that the Universe was smaller in the past and thus denser with a higher temperature for the CMBR. Based on observed red shifts of distant galaxies, we can then deduce, but not directly measure, what the CMBR temperature was at each distance.
What the authors of the above paper did was make a direct measurement of the CMBR's temperature at a specific time in the past. The measured temperature is higher than it is today which implies a denser and thus smaller Universe. The researchers further found that the directly-measured temperature fit neatly with that inferred from the observed red shift of the galaxy being studied.
In a nutshell, the chain of inference is swapped:
These two inference chains based on different sets of evidence neatly complement and support one-another.
One thing to note is that the CMBR wasn't created by expansion (at least not directly) rather it's expansion that explains its current temperature and uniformity. Per the Big Bang theory, the early universe was very dense; so dense and hot that all matter was a plasma of subatomic particles, opaque to photons. At about 380,000 years after the Big Bang the Universe had cooled (through expansion) enough that protons and electrons could combine to form neutral Hydrogen gas (which is transparent). The CMBR is the light that was set free at this time and has been cooling ever since.
But is it still possible that the redshift is caused by some unknown phenomena and not by galaxies moving away from each other?
In history some alternative theories were proposed, like the tired light hypothesis, the steady state universe etc. But the observation ruled these and other theories out.
See also Alternative cosmology
There are no other reasonably direct methods, but there are definitely indirect methods. One, in @Alex Hajnal's answer, the higher CMB temperatures measured further out is a very nice indirect measure.
Another indirect piece of evidence, which no one has noted yet, is that as we look further and further out, the universe looks younger and younger, and less and less like what we see in our neighborhood. You are pretty much forced to explain that scientifically by saying that the universe had a beginning on the rough order of 10 billion years ago, and that stars and galaxies only started forming then. (This isn't proof of a Big Bang specifically, but it does eliminate most alternatives to it. The Steady State model, for example, is falsified.) It's very very hard to explain what we see except as being due to a universe expanding from a hot dense state ca. 1010 years ago.
More indirect evidence comes from General Relativity, a theory of space, time and gravity which is very well verified -- it's been tested for a century now and challenged by countless other theories, and only GR has passed all experimental tests. GR robustly predicts that a static universe is impossible and that it must either be expanding or contracting. This is indirect evidence from mostly local experiments.
Yet more indirect evidence comes from nucelosynthesis calculations which show that the H/He/Li ratios that we observe in the oldest and least evolved stars is exactly what we predict based on applying the measured properties of nuclei to a Big Ban fireball.
There's so much science other than the red shifts which point to the universe expanding from an initial very hot, dense state that even without the observation of red shifts, we'd eventually be forced to that conclusion.
In addition to the circumstantial evidence provided by the other answers, a strong verification of galaxies moving away from each other is given by the fact that we see physical processes — such as the decline time for the brightness of supernovae — increase, the farther away it is. For a source with a redshift of $z$, the amount of this time dilation is observed to be $(1+z)$, exactly in accordance with what is expected from general relativity in an expanding universe.
That is, a supernova observed with a redshift of $1$ takes twice the time to decline as a local supernova.
Note though that this is not a verification of the expanding Universe, only of galaxies moving away from each other. If the Universe were static, but the galaxies moved through space, you would observe the processes dilated by the same factor, as predicted by special relativity. There are, however, other evidence that the galaxies do not move through a static space, but instead lie more or less still in an expanding space.
Yes:
The important thing is that these results not only say the same, but they are also corresponding eachother.
OK, this answer involves red-shifts but hear me out.
Under General Relativity, multiple mechanisms can create red-shifts: expansion of space, objects moving relative to an observer (i.e. us), and light moving out of gravity well. The latter option is outside the scope of this question and the former is excluded from consideration at the questioner's request. That leaves only the second option (relative motion, a.k.a. the relativistic Doppler effect) under consideration; this shifting can be (and has been) tested here on Earth and has been shown to exist.
Red-shift is observed in all apparently distant objects (dim, low metallicity, etc.). From the red-shift of the spectra seen in any given object we can determine how quickly it is moving away from us. For example, an object with a measured red-shift of $z=0.5$ is moving away from us at about half the speed of light. So far so good. The trouble arises when we observe objects with $z>1$. Many such objects have been found; the current record-holder is GN-z11 with a red-shift of $z=11.09$. Put another way, if only relativistic shifting were at play this object would be travelling away from us at over 11 times the speed of light.
Given that no object having mass can reach light-speed it is clear that the observed red-shifts cannot be caused by relativistic motion. Since there are no known mechanisms beyond the three listed above that can cause red-shifts in the spectra (compare extinction), the only explanation matching these observations is the expansion of space. Put succinctly, the fact that superluminal red-shifts are observed at all is evidence that space is expanding.
