A record-breaking achievement is better than any previous attempt by anyone, or at least better than any officially documented one. If you solve the Rubik's Cube faster than the fastest solver, that would be a record-breaking feat!
When someone sets a record, they accomplish something that hasn't ever been topped: no one has run faster, or eaten more hot dogs, or lived longer. There are also records for things like the hottest day, fattest bear, and most powerful telescope. Eventually, nearly every record gets broken, and the event that does it is record-breaking. So if you practice every day, maybe you can one day be a record-breaking Rubik's Cube solver — but you'll probably have to do it in less than three seconds!