What's this white strip on the top side of my boneless skinless chicken breast?
Structurally it is similar to silver-skin on tenderloin and other red meat muscles, some connective tissue marking the muscle boundary.  I have seen it be a little tough on older, non-broiler birds but even then not normally worth messing with, IMO.  If you chunk the meat you might find spots that are thick enough to be worth removing especially near the shoulder joint, but otherwise probably not.