It means to play very hard and without reservation. So, a person who "plays his socks off" put 100% of his energy into the game, and has no energy left after the game. For some reason, English has a lot of idioms about losing one's socks. - "knock socks off" (big surprise, or a brutal beating) - "scare socks off" (big scare) - "work socks off" (work really hard) These idioms do not make any sense. It's just something that we say.

"He played his socks off" simply means "he played extremely hard." Nothing else is implied.

Just imagine someone running so hard his socks come off. That is exactly what this idiom means.

The idiom derived from "knock socks off" Usually means something is done or completed in an extreme way, or a great degree. "He played his socks off" means that "He has played the game in a very extreme way, or great degree (regardless of outcome). "