Why do IR remotes influence AM radios?

This IR signal is indeed ignored by the AM radio. However, an AM radio is very sensitive the radio waves (yeah, DUH! ;-) )

When the IR remote operates (you push a button) the chip in the remote will switch on a clock resonator circuit which it needs to generate the IR signals. I have seen most IR remotes using a 455 kHz resonator. This is simply used because it is cheap.

The IR remote chip has a circuit to divide down this frequency to get the 38 kHz it needs. A division by a factor 12 would do as 455 kHz / 12 = 37.9 kHz. Yes that is "close enough" as the IR receivers aren't that accurate, they cannot distinguish between 38 kHz and 37.9 kHz. Also, that isn't needed, the 38 kHz is just a "carrier" it does not contain information.

So we now have 38 kHz which is a signal that has a square wave shape when it comes out of the IR remote chip. This is because this is simple (logic circuitry works with square wave signals) and the IR LED needs to be on or off. So there is no need for "in between" levels.

Now a property of a square wave signal is that it does not only contain a single frequency (like 38 kHz), it also contains many multiples (mostly uneven harmonics) of that frequency as well so: 2 x 38 kHz = 76 kHz, 3 x 38 kHz = 114 kHz, ... 14 x 38 kHz = 532 kHz. There you go, the 14th harmonic is already on a frequency the AM radio can receive!

Never underestimate the harmonic content of switching and square wave signals. I once worked on a product where the 238th harmonic of a DCDC converter running at 600 kHz was disturbing the receiver which was working at 142.8 MHz!

Most likely, your radio is picking up unintended EM radiation from the remote's circuitry. You mention that it operates between 30 and 38KHz, but the IR probably uses square wave modulation, so you'll still pick up the harmonics. Of course, it could be some other signal than the LED drive getting picked up.

Once you have a signal or harmonic near the frequency your radio is tuned to, the radio will heterodyne it down into the audio band. Try it with a calculator, those can be even more amusing if you have a noisy one.