why do pro athletes appeal their suspensions even though what they did was wrong?

Answer 1

The same reason people hire a lawyer after they commit a crime.

There is always a chance that something can get overturned on a legal technicality.

Also, when appealing, the suspension doesn;t start, so you can still play what could be important games, especially in the NFL where the season is short and a lot of games are important

Answer 2

Appealing gives you the room to move the suspension around. Say you appeal, and a hearing is called for a week. A player gets to play an important game, and then drops the appeal and serves the penalty.

There are many cases of punishments being lessened, of course, and that's why there is the appeal process. Sometimes 2 games can be much better than 4.