The player simply wants to check his hole cards to see if the card he drew helped him.
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When this player catches the small card and double-checks, you can be pretty sure he has some other small card in the hole. That small card did not have relevance until now. Why? Probably because he had a pair of aces to begin with and some secondary card that was not worth remembering. Catching the three made him check to see if he caught two pair. If both his face-up cards were suited, a motive for a double check could be to see if he has one more of that suit in the hole.
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In selecting your strategy, figure that there's a better than average chance that his opponent has a pair of aces and a fair chance he has aces-up. There is almost no chance he has either three aces or three threes. And it's very unlikely that he has a high kicker to go with a pair of aces. Keep these factors in mind, consider your own hand and play accordingly.
