選ぶ means 'to choose' and 手 means 'hand' or by extension 'ability'. I don't see how this can yield 'athlete'. Is 選 ateji?
1 Answer
A dictionary definition of 選手 is
スポーツで選{えら}ばれて競技{きょうぎ}に出場{しゅつじょう}する人
A person chosen to appear in a sports competition
So, 選 expresses 選ばれる, and 手 expresses the person in question.
You noted that 手 means "hand," and by extension "ability," but it's also, by even further extension, the person who possesses that ability.
You can see this in words like 運転手{うんてんしゅ} (driver) and 歌手{かしゅ} (singer) as well.