First introduced by Iron Woody, the band-assisited pull-up simply allows you to do more reps than you normally can. More reps means quicker muscle development. As your strength increases, simple switch to a lighter band, until you don’t need any band-assist at all.
Incorporating bands into your lifts increases resistance at the end of lift, where you need it. They also introduce a slight element of instability during the lift which helps develop adjacent muscle strength.
It has been proven that incorporating bands into many of the Crossfit routines will dramatically increase your results. Ask your Crossfit trainer to show you how to do it properly and safely, and see the improvement in your workout results.