A CD is simply a digitized version of those vibrations, called a wave. If you truly understand how a CD works, a record is basically a much simpler subset of that process: No conversions from/to digital to/from analog. The waveform of the audio is stored exactly as a physical reproduction of that waveform.
A cassette tape works on the same principle but instead of the vibrations being stored as a physical groove in a record, it is stored by magnetizing those vibrations onto a metallic strip of material over the length of the material. Playing it back is somewhat like placing a tiny piece of metal by the tape as it moves and as the tape goes by that little piece of metal, the magnetic changes that were stored on the tape move the little piece of metal. Those movements/vibrations are amplified and that's your audio.
These are very simplified explanations, but only to get the ideas across.
It isn't magic.