Electronic – Why do AM radio stations have secondary frequencies

Your mention of the frequency (97.5 MHz) tells us this is an FM receiver. (AM will behave differently, as will other modulation schemes).

Because FM is encoded by modulating the signal frequency, anything to do with AM is undesirable. To deal with this, most receivers over-amplify the signal until it becomes larger than the later stages can pass. The signal then "clips" to the voltage of that amplifier. This stage is called a "limiter"--it limits the amplitude to some fixed value. In theory, any signal weaker than that drops out and just becomes noise, and any signal stronger than that has a very nice fixed level that the FM detector can handle without having to worry about amplitude variations.

The amplifier-limiter stages create a phenomenon called "capture", where the strong signal tends to eliminate the weaker one. This is why you hear only one station.

If the signals were very close in strength, you would indeed hear them "mixing together", but that only happens for a fraction of a second as the signal levels rapidly change (presumably, you are in a vehicle), so you normally don't hear that.

Radio frequencies from AM stations get into audio amplifiers not only through the inputs, but also via the outputs.

The speaker cabling can serve as an antenna to pick up radio signals. This is is coupled back to the output of the amplifier. But the amplifier has a negative feedback loop which feeds the output back to the input. So the output is really just another input.

Production amplifiers usually include a Boucherot Cell on the output, often an output inductor after that. Both of these devices can help block incoming RF, even though it's not their main function. The Boucherot cell is simply a capacitor in series with a resistor, placed between the output and ground. Common values are 0.1uF and 10 ohms. You can see this in many amplifier schematics. Edit: I see there is a 10 ohm/0.047uF cell in the amplifier; did you install that in the built circuit?

When you say "nothing connected", of course you have the speaker connected, which is how you hear the radio stations!

There is also the possibility that the circuit itself is picking up interference. There is a reason why amplifiers are built built into metal boxes and why fuss is made over any internal wiring and in particular grounding! If your power amp is just bare components on a breadboard, don't be surprised if it is susceptible. Radio waves are falling on your nest of wires and components. You have high gain in your circuit, and nonlinearities, so these oscillations are amplified and rectified.

There is also the power supply connection. A single-supply amplifier like the LM386 is very susceptible to noise coming in over the power supply. You have to bypass the power supply very well. A single large capacitor may not be enough; you need a small 0.1 uF ceramic close to the IC power pins.