The simple answer is that light from the distant parts of the Milky Way Galaxy have already been traveling for that amount of time. Because light does not travel instantaneously through space, we are seeing the galaxy not as it is now, but as it was 100,000 (or so) years ago.
Incidentally, this is also how we are able to see back to earlier moments of the universe's history. In terms of distance, the further we look, the longer light would have had to leave in order to reach us today.
The universe did have a beginning (at least in the sense of having an earliest time). Because the universe is not infinitely old, there is actually a limit to how far we can see back into the universe's history (see: Cosmological horizon). We are able to make maps of the universe, like this one from SDSS:
by running what's known as a survey. These are large projects which maps out the positions (and potentially other properties, depending on what their science goals are) of galaxies in the universe. How you should look at this picture is the following: 1) You're at the center of the image, 2) Each point in the picture represents an individual galaxy, 3) Distance from the observer, the redshift of a galaxy, increases with increasing distance from the center, and 4) as you travel around the circle in angle, you're sweeping through the right ascension coordinate of the celestial coordinate system (think longitude). You're seeing a pizza slice instead of a sphere because this is a particular slice in declination (lattitude). As you look to higher redshifts, you are looking further back into time, and earlier into the universe's history. This is how astronomers/cosmologists can say something statistically about models of structure formation of the universe (how objects like galaxies, and clusters of galaxies came to be).
The problem with looking at galaxies at very high redshifts is that they become
very faint. At a certain point, you need more powerful telescopes to continue looking further back in time. Though, we can get around this somewhat by looking at the radiation component of the universe at microwave wavelengths (most surveys are in the optical region of the electro-magnetic spectrum). This radiation is called the cosmic microwave background radiation (CMB), and can also be mapped out (see: COBE,WMAP, and most recently Planck satellites).
This is also a map of sorts. This map of low energy photons is a snapshot of the universe approximately ~380,000 (redshift of z=1100; meaning the universe was smaller by about a factor of 1101) years after the big bang, when the universe had very little of what we would recognize as structure. The job of the cosmologist is to connect the picture we see from the CMB to the picture we see from our surveys of galaxies through the use of physics.