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Narrators of astrophysics documentaries constantly state that matter spirals into black holes as if it were intuitive that this would happen because of gravity. But the moon isn't spiralling into the earth. Newtonian physics suggest that an object would either fall past the event horizon in less than half a revolution, or it would orbit indefinitely until acted upon by a third party.
Why would an orbit gradually degrade?
There are probably four different processes, any or all of which may be described as "spiralling into a black hole"
The effects of general relativity. as @AtmosphericPrisonEscape has indicated, once you get very close to a Black hole Newtonian physics ceases to be a good approximation and must be replaced by general relativity (GR). When you do the calculations, you find that all circular orbits within 3 Schwarzschild radii of the black hole are unstable. Any object orbiting closer than that will definitely spiral in.
Gravitational radiation GR tells us that any two orbiting objects will emit gravitational waves, which carry away energy and angular momentum, slowly reducing the separation between the objects. This effect is, however, incredibly small, unless the objects are very massive and orbiting very close to one another (which means they must also be very dense). Thus this is mostly relevant for black holes and neutron stars in close orbits.
Turbulence in an accretion disk When a lot of gas or dust (which may be the remnant of larger objects pulled apart by tides) is orbiting a black hole, it tends to form a disk, called an "accretion disk". Basically the disk is oriented according to the average angular momentum of the stuff falling in, and anything orbiting in a different plane will collide with lots of the material in the disk and quickly join the disk. This is the same mechanism that caused the early proto solar system to form a disk, and indeed Saturn's rings (I know I'm oversimplifying somewhat). Anyway, to remain in orbit, the inner parts of such a disk must be rotating faster than the outer parts, and this "shear" in the flow of gas tends to cause turbulence, which, it turns out, tends to transfer angular momentum outwards through the disk. This results in material at the inner edge of the disk getting slowly closer to the Black Hole until it reaches the Innermost Stable Circular orbit, at which point effect number 1 takes over.
Dynamical "friction" If there are a lot of larger bodies (stars, asteroids, etc.) orbiting a large central mass, they will, from time to time, pass near each other, and change each others orbits, transferring angular momentum between them. It turns out that averaged over many encounters, the effect of this is to drive larger bodies inwards and smaller ones outwards (possibly ejecting the smaller bodies from the system completely) so that it acts as something like friction on the larger bodies.
The moon isn't losing energy by its interaction with the rest of the solar system. On the contrary, it is gaining energy from its gravitational interaction with the Earth, while the Earth itself is gradually losing rotational energy and transferring it to the moon. The moon therefore is slowly moving away from us at the rate of several centimetres per year. The accretion disc orbiting a black hole, on the other hand, is losing energy through radiation caused by friction and compression, so the particles in the disc are gradually falling into the event horizon.
Why would an orbit gradually degrade?
With planetary orbits being stable around a star for long times, one would think that the change of angular momentum (its usually angular momentum, not energy that you want to talk about for orbital decay) for celestial objects must be about zero.
In fact it is more the rule than the exception, that an objects orbital angular momentum changes with time. Early in their history, the giant planets in our solar system migrated quite significantly (for which there is more and more evidence), and as you stated yourself, the Moon also recedes from us.
There is a fundamental reason why this can happen, and it is the same reason why things must inspiral into a black hole.
First things first, I have to define a bit of math. A potential $\Phi(x,y,z)$ is a scalar function, whose 3D-derivative gives a force vector $\vec F(x,y,z) = \nabla \Phi(x,y,z)$.
If you're unfamiliar with this notation, then the key message here is: A potential contains essentially all information that a conservative (!) force does, and physicists use it all the time because handling one scalar is easier than handling three scalars (which is the force vector).
Then, without proof, I state a theorem from classical mechanics, namely that the only force laws that can produce closed orbits are potentials of the form $\Phi\sim r^2$ and $\Phi \sim 1/r$. The former is a Hooke spring-like law, the latter is a perfect two-body gravitational potential.
Tides and perturbations from the planets are relatively weak contributions compared to central gravity. In the case of a black hole however, you pick up an additional strong-gravity potential term $\Phi_{\rm BH}\sim 1/r^3$ which has a much stronger radial dependency than your regular gravity term, and is purely attractive.
Because of this, black holes posses a so called innermost stable circular orbit (ISCO), beyond which nothing can orbit the black hole without drifting radially inward. Because one can show that ISCO and the black hole's Schwarzschild radius are related via $r_{\rm ISCO} = 3 r_{\rm s}$, one will never encounter the ISCO in regular solar system dynamics.
Any orbit requires velocity to maintain. If the moon's orbital velocity slowed its orbit it would come closer to Earth. If it were to speed up it would move further away. If the moon were to stop in it's orbit it would immediately fall like a giant stone.
Inside a black hole a velocity greater than light is necessary just to maintain a stable orbit. Since such velocity is impossible then the only possibility is a decaying orbit ("spiraling in") that inevitably leads to collision with the singularity.
