If you egg someone on, you are inciting or encouraging them to do something that usually isn’t good.
“I didn’t want to do it, but my friends were egging me on and said I was too scared.”
“Stop egging on your brother to do stupid things, you’re the one who will get into trouble.”
“to egg” (without the preposition) means to throw an egg at someone, usually a politician, to protest against their politics or behaviour:
“The prime minister was egged by an angry protester during his visit to the factory.”