Why does Avogadro's Law hold true for ideal gases?
Avogadro's Law actually states that the volumes of ideal gases at a fixed temperature and pressure are proportional to the number of molecules that are present. This is actually a sub-case of the more general ideal gas law, which states $PV = nRT$ (or $PV = nkT$, if you're working on a per-particle basis) .

The ideal gas law actually falls out of a basic statistical mechanics analysis of the behavior of a collection of molecules which interact solely by elastic collision. (Here's a 25 minute video of one such derivation.)

Basically, the more particles you have in a given region with a certain fixed speed (i.e. a fixed temperature), the more frequently they collide - not only with each other, but with the walls of the container. This means that putting more particles in the (same size) container (at the same temperature) increases the outward pressure on the container walls. In fact, the rate of collisions (and thus the pressure) is directly proportional to the number of particles (doubling the number of particles at a fixed temperature doubles the rate of collisions). 

But we want to hold the pressure constant! If we can't change the temperature, one way to reduce the pressure is to increase the volume (e.g. let the container expand like a balloon.) As the volume grows, each particle has to travel farther before it can collide with the walls of the container, meaning that the rate of collisions and thus the pressure on the walls gets less and less. 

It just so happens that this relationship between pressure and volume is inversely proportional (that is, Boyle's Law). So to halve the pressure, we need to double the volume. So if we double the number of particles, to keep the same pressure we need to also double the volume. Or more generally, the volume is increased in direct proportion to the number of particles - that is, each mole of gas takes up a certain volume (at a fixed temperature and pressure). If we try and put it into a different volume, the pressure (or the temperature) will change.

Okay, but why doesn't a fixed number of different particles take up different volumes? For example, why doesn't doubling the mass of the particles make it take up half the space (or double, or a quarter ...)? - Heavier particles might hit the walls of the container harder, but at any given temperature heavier particles are moving slower then lighter ones, so the collision rate is less. When you run the numbers and calculate the velocity of the particles, the rates of collision, and then how hard they hit the container walls, the mass cancels out. At a fixed temperature, the contribution to pressure from a single particle colliding with the walls of the container is independent of its mass.

You can do the same sort of analysis with other properties. They are either 1) are explicitly discounted by the assumption of "ideal gas" (e.g. no particle interactions) 2) don't factor into the calculation at all (e.g the absorption spectra of the atoms) or 3) precisely balance out - leaving you with $PV = nRT$ being the relation for an ideal gas.