This phrase means that he played even though he had pain.
"Through" in this context is referring to the idea that he didn't let the pain stop him or slow him down, so he played right on through it.
This is just a common idiom that I think most native English speakers never stop to analyze. When they grew up they heard the phrase at some point and gathered the meaning from the context, and it just became part of their vocabulary - without really analyzing whether it even makes sense.
In sports, many somewhat nonsensical phrases have evolved from overly excited sports announcers, etc. Many of those phrases make far less sense than this one, but they continue to propagate since they're funny or rhyme, etc.
Note: here are some more interesting sports phrases:
He schooled him - He completely defeated his opponent and made him look bad while doing it.
He got schooled - The opposite of the previous phrase; he lost very badly.
He got thumped - Again, he lost very badly.
He got his butt kicked - He lost badly.
He belted that one - He hit the ball very hard.
Grab some pine - You played poorly, go sit on the bench and let someone better play
He grew roots - He didn't move when he should have
Nothing but net - Great basketball shot where it didn't even touch the rim, it just went perfectly through the hoop.
Note: many of these are considered slightly offensive, but are intended to antagonize your opponent during a match.