The word 对手 is formed by combining the character 对, which carries the core idea of facing something, opposing it, or being its counterpart, with the character 手, which literally means "hand" but here is used metaphorically to mean a person (as in someone who performs an action or has a skill). Together, they create the concept of a person who stands facing you, implying direct opposition, rivalry, or an equal match in a contest or game.