They work by sending a radio signal which is "modulated" by the voice of the sender. The simplest kind is called amplitude modulation. The circuit generates a constant frequency signal at, for example, 300MHz (300 million cycles per second). Think of a tuning fork, but at a much higher frequency. Then the amplitude (analogous to power, or volume) is made the same as the amplitude of the signal coming from the microphone. The human hearing range is only up to 20 kHz (20 thousand cycles per second) which is much slower than the radio signal, so it is easy to modulate the radio signal with the audio signal. Then the receiver homes in on the transmitter's frequency, filters out other radio signals, and detects the amplitude of the signal at that frequency. Then this is sent to the speaker. This is how the very first radio transmitters worked using the very simple crystal rectifier detectors which were available at the time. It is also how AM radio stations work and how early walkie talkies worked.

Now with more advance circuits available using transistors and digital processing more advanced types of modulation can be used. FM radio changes the frequency of the transmitters signal by an amount controlled by the audio signal. This requires a more complex circuit to convert the radio signal back into an audio signal but it provides a better quality transmission which is why FM radio sounds better than AM radio. Typical walkie-talkies in the US now use FM signals.

Walkie-talkies are radios. The circuit inside the walkie-talkie runs electrical current through an antenna to broadcast radio signals, and when listening, the antenna picks up radio signals because radio waves induce electrical current to flow in an antenna.