He told me he plays table tennis well. Why didn't he use played?
The simple present is used for general statements. They might be specific to the time of speaking, or they might be much broader. If I say "I speak French" (though I don't, at least not very well), I am not saying that I am speaking French at that moment. I am saying that I have the ability to speak French, or that I may speak French from time to time. If I say "I speak French poorly", I am saying that, on those occasions that I speak French, I do so poorly.
"I spoke French poorly" would be used if either I no longer speak French, but when I did I spoke it poorly, or in reference to a particular occasion or point in my life.
"I play table tennis well" fits this same pattern. It would mean that, on those occasions when I play table tennis, I play it well. "I played table tennis well" would indicate that I used to play it well, but I do not play it any more, or that on some particular occasion in the past (made clear in context) I had played it well.
 I play table tennis well. [original utterance]
 He told me he plays / played table tennis well. [indirect reported speech]
He could have used "played".
This is all about something called 'backshift'. Backshift with verbs of reporting that are in the preterite is perfectly normal.
The use of a non-backshifted report ("plays") is possible but subject to the condition that it must be reasonable to assume that the original utterance is still applicable and relevant.
Only if this condition is satisfied are both backshifted and non-backshifted versions in principal possible.
Thus in your example, which uses "plays", we have to assume that he still plays tennis well.