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"I have fun": I enjoy myself, I have a good time, etc. Examples:
I have fun playing video games. I don't have a lot of fun in math class. I think she'll have fun on the roller coaster.

"I am fun": People have fun when they are around me, I am spontaneous, I am extroverted, etc. Examples:
People say I am a lot of fun. She's fun (this is sometimes a backhanded compliment).

In general I would not use the second phrase unless you're very comfortable with it, it has a lot of connotations and can sometimes be an insult if used in a particular way.