How complicated is the heating of meteors? When any gas, including our atmosphere, is compressed, it's temperature rises.
When a meteor is falling through the atmosphere, it is compressing the air below it.
Because this makes the air hotter than the meteor, heat moves from the air to the meteor.

Also, some amount of oxidation (burning) might also take place.

What I would like to know, is it really that simple?

Alternatively, what else causes meteors to heat up?

I've heard people say "air friction," which sounds to me like nonsense, but I am not a rocket scientist, so what do I know?  If "air friction" is real I want to know how it works!

Also, I'm fairly certain that heating of air due to turbulence will only affect the air which the meteor has already moved through... not the meteor itself.  If I'm wrong, I want to know!
Your analysis is correct. Compression of the air is the cause of heating. 

You call this "simple" but the compression depends on the ability of the air to flow around the meteoroid, and as the modelling the hypersonic flow is far from trivial.

"Air fiction" is the resistance to motion felt by an object moving relative to the air. That includes the reaction to the force that is compressing the air in front of the meteoroid, but also results from work done in moving the air out the way, and is dependent on shape, surface effects, turbulence etc.

The visible meteor is the trail of plasma produced in the air by the meteoroid. The actual piece of rock is too small and dim to be seen. The meteor is glowing air.