Vinyl records have grooves in them. These grooves have bumps and bends that correspond to the recorded music and voices.
As the record turns, the tip of the pin travels within these grooves, following these bumps and bends and causing the pin/needle and cone/cup speaker to vibrate.
The vibrations in the cone/cup cause the air to vibrate. These vibrations then come out from the cone to your ear, which you hear as music, voices, or whatever has been recorded.