The use of the "winning" condition is to ensure that the overall construction has the properties we want. For example, in the game in the original question, Player 1 chooses "left" or "right" at each stage. Thus Player I always has a possible move. But Player II is forced to lose if the play arrives at a neighborhood that contains only a single point. So the actual question is whether Player II can play in a way that avoids ever being backed into a corner like that.
A winning strategy for Player II in that game actually encodes a complicated dependent-choice construction in which none of the neighborhoods played ever contains just one point. Such a construction is not possible in every Polish space. But we know that the construction is possible if and only if player II has a winning strategy in that game. So the game-theoretic language gives us a way to characterize topological properties of the space.