Why do jockeys stand up in the stirrups after crossing the finish line?

Standing up tells the horse to slow down and that the race is over. It's also easier on the rider to slow the horse from a standing position, because it gives him more leverage. Every jockey, and all exercise riders do this. Even riders in non-racing sports, such as eventing, will do this from time to time. It doesn't hurt the horse, and it reduces the chance of an accident.

Without going into too much! Standing up at the end of the race is a safeguard cue move, as this helps to get the horse to slow down. In other words this stops the horse from running. It's kind of like a suspension system.

All the jockeys - not just the winners - straighten their legs up while keeping their hands down once the race is done.  Riding in the race position requires a lot of leg and core strength, and is very hard on the knees and back.  Once the race is over, they shift to an easier position.  

This does not in anyway hurt the horse, and it cues him that the race is over and he can slow down and ease up.

Even when the race is won and the jockey is "celebrating," it doesn't hurt the horse.

Celebrating..... and I hate to see that - from the horse's perspective.