The expression does not mean that he played until the pain ended.
It is understandable that you should be confused in this particular case: pain is a state and play is an activity, both of which have duration, so it is natural that you should think of through as having the sense of throughout—from the start of the pain to its end.
We sat through a long and boring lecture.
He played at the top of his form through the entire game.
But in fact through here is used in its ‘primary’ sense of ‘in one side and out the other’, as when we speak of breaking through a barrier. Pain presented a (continuous) obstacle to his playing, but he successfully (and continuously) ‘penetrated’ the obstacle and played (continuously) anyway.
Here are a couple more examples of penetrative through in a durative context:
We drove through a blinding snowstorm to Prague.
He maintained his high purpose through political opposition and bureaucratic delay.