I am not very familiar with physics or rocketry and I am having a lot of difficulty grasping why spinning a rocket or satellite stabilizes it. Most sources I have looked at have said either that it works due do gyroscopic effects or because the spin averages out forces over a period of time, but I can not seem to understand why.
Conservation laws may never be intuitive; what makes them laws is simply that after all this time we've never seen exceptions to them or the things that imply them.
Hugh? In a given scenario? you ask? If the rate of spin is insufficient, or you've unfortunately chosen the intermediate axis to spin around, this can fail. For more on the intermediate axis theorem see answers to Does the “Tennis Racket Theorem” apply to the ISS? Does it rotate around its intermediate (unstable) axis?
Spin stabilization is not perfect, it just keeps one of the the spacecraft's axis from wandering more than a small angle away from its initial direction, because nothing is expected to add anywhere near as much angular momentum as you've just given it. It will however cause it to precess, and that's just a result or consequence of conservation of angular momentum and the math of rigid body dynamics. To most people, those are just not intuitive.
Each person intuits differently, here's as far as I've been able to intuit this.
If you put a spacecraft in empty space at a certain attitude with absolutely zero rotation and don't touch it, it will probably continue to keep that attitude.
In the real world there are always torques from uncentered thermal radiation by the object or light hitting it or gravitational gradients or magnetic fields or their gradients if there are magnetic materials or electrical currents on the spacecraft or very slow leaks or outgassing or differential drag even in the "vacuum" of space or...
It will start turning. Even if it's tiny, over several years (let's say a trip from Earth to an outer planet) it will turn and point in a new direction.
However if you add a small, definite angular momentum to the object, even a few rotations per minute, it's going to be absolutely huge compared to the rotations that those sources can produce. Nothing that one can expect the spacecraft to encounter will contribute more than a tiny amount of angular momentum compared to this.
Given that size difference, we can count on the axis of the spacecraft continuing to point almost exactly in the same direction for decades, assuming we spun it fast enough and designed out anything that could add a similarly large spin.
If it does acquire a change in angular momentum, this will just cause it to precess slightly. Spin stabilization is not perfect, it just keeps one of the the spacecraft's axis from wandering more than a small angle away from its initial direction.
Precession. But if precession is intuitive to some it certainly is not to me. It's just a result or consequence of conservation of angular momentum and the math of rigid body dynamics.
The most intuitve way to explain spin stabilization would be a spinning top, e.g. this one:
You can imagine the satellite as the spinning top, as long it is spinning, it does not fall but keeps a quite stable position.
Now explaining how a spinning top works is more complicated. In short because of the Angular momentum "saved" in the spinning top conteract applied (disturbance) forces with a momentum... and you may see that explaination ist everything but intuitive.
There is an idea (BUT THAT IS NOT CORRECT SPINNING TOP PHYSICS!!!) of thinking about it this way: imagine a spinning axis. now some kind of force is trying to tilt it in a direction, after a half spin this force would be in de exact oposing direction and the effect of this force would be 0 over a whole rotation.