Indulge is a verb that means "to give in to something," like when you indulge your craving for chocolate by eating a big piece of it.
The original meaning of indulge is "to treat with unearned favor." That's why it doesn't mean the same as reward. Instead, when you indulge, you enjoy or take part in something just because you want to — like indulging in a celebrity gossip magazine: you know you should be doing something more constructive, but you don't care. When you indulge someone else, you go along with whatever they want to do.