Your friend did something funny, so you started laughing. She's mad because she thinks that you're making fun of her. You didn't mean to make fun of her, so you say:
No, come on! I'm not laughing at you; I'm laughing with you.
"Come on" is something to say when you're trying to persuade someone:
You also say "Come on!" when someone is upset or angry, and you're trying to get them to calm down:
Come on. Don't talk that way. Of course I love you.
"Laughing at" someone suggests a mean, hurtful attitude. You're enjoying someone's embarassment or making fun of their weaknesses. Of course, you can "laugh at" a joke or "laugh at" a funny movie without it being mean, but "laughing at" a person isn't nice.
When you're laughing because of something that a person did or said, but you don't intend to be mean, you can say "I'm laughing with you, not at you." "Laughing with" someone suggests that you're both laughing and having fun.
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