Do this: Stretch a rubber band between the thumb and a finger of one hand. Then pluck the two strands gently. They wiggle back and forth very fast. (Víbrate is the correct word for the wiggling.) When the strands víbrate they make the air víbrate, too, and so you hear a sort of purring noise.
Your cat has little stretchable bands in its throat. These aren't made of rubber, of course. They are a special living part of the cat's body. When a cat feels relaxed and satisfied, it can make the bands víbrate in its throat. As the cat breathes, the air picks up the vibrations, and so the sound of purring is carried through the air to your ears.