These players, the fan favorites, are favorites because they realize that they are ambassadors of the game, and many will
show up at the game even if they are unable to play
What is the most likely reason the author includes the phrase "the
fan favorites" in the sentence?

To express the value of the players cause they are the most important players according to the fans

That would be a run on sentence. A sentence fragment is something like: was doing, when he was doing would be the full sentence, was doing being the fragment. And a run on sentence is exactly what it's called, a sentence that goes on and on, most of the time with little commas and periods. Hope this helps!