What is a swivel?
A swivel is something fishermen tie to their line so it doesn't get too twisted. Basically, it's a ball-and-socket joint, so it can rotate when the string gets twisted.

How do they apply to yo-yos?
Swivels can maintain a neutral twist in yo-yo string just like they can keep a fishing line straight. They are most useful when doing large numbers of loops or similar looping tricks that twist the string one way. Cut the string, tie one end to one side of the swivel, and the other end to the other side. You want to have the swivel as close to the finger loop as possible, so it doesn't get into the gap of the yo-yo. Then, you should be able to do unlimited loops until your arm removes itself from your body.