How does Frequency Modulation work with radio frequencies?
Your FM radio works because it receives a band of frequencies around the frequency it shows or that you tune it at.  As you say, the nature of FM means that the frequency will vary.  However, the extent of these variations is well defined, and the radio is designed to "find" the carrier anywhere in this range.

This may surprise you, but AM also requires the receiver to pick up a range.  I'm not going to go into Fourier analisys right now, but basically changing amplitude is adding frequencies.  Put another way, a true pure single frequency can't ever change in amplitude and can't carry any information.

The way AM works, there is a frequency band on either side of the carrier that is the width of the highest frequency that can be transmitted.  For example, if the signal being modulated onto the AM carrier can be up to 10 kHz, then there is a 10 kHz band of stuff on either side of the carrier.  In fact, the carrier is constant and the actual information is in these side bands.  Yes, I know that may be unintuitive, but at the level you are asking and what I have time to explain here you'll just have to trust me on this.  Look up Fourier analisys if you want to learn more.

For example a AM radio station a 1 MHz carrying up to 10 kHz content will have a signal spread out over the range of 990 kHz to 1.01 MHz.