Microphones and cameras have sensors that react to vibrations in the air and to light, generating an electrical signal in response. The device then measures that signal and notes it down over time. In analog devices, this was done by transcribing it as a magnetic signal onto a tape. In digital devices this done by measuring it as a very precise number, and storing those to memory of some sort.
To play back the sound or the video, a device takes the previously recorded measurements and converts them back to electrical signals, which get sent to either a speaker or to pixels on a screen, which then either vibrate the air or shine with a strength proportional to the original signal, recreating the original sound or picture.
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