Why is my audio amp picking up AM and FM radio stations?

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.

An FM signal will Amplitude Modulate a tuned circuit. A non-linear circuit will do peak-detection on an AM signal. An Audio Amplifier will amplify small signals detected by peak-detection of AM signals.

You prevent that by adding low-pass capacitor-resistor filters to your inputs, and high-pass filters to negative feedback paths.