What is the best way to clear up canker sores caused by braces?

Wax! Most oral sores associated with braces are traumatic in origin. Wax on the offending wire will do the trick.
No simple answer! There isn't one remedy or simple answer to this. For some, it is a simple matter of doing a better job brushing and flossing. And yes, it is much more difficult and time consuming to do a good job of it while undergoing orthodontics. Unfortunately for some, it doesn't get any better no matter what until the braces come off at the end of treatment.