I read a comic and a character uses this word when the day is over and the other character said he experienced something unusual. So what's the meaning of try me? What is the something to try? Why me?
The phrase "try me" is used to suggest that someone may be willing to do something unexpected or unlikely (OED):
You won't use the gun here. - Try me (meaning: Oh, yes I will. If you don't believe me and think that it's unlikely, I'll show you that I can use the gun here).
You probably don't know the answer. - Try me (meaning: I might know the answer; give me a chance to answer the question).
I guess you don't know who this man is. - Try me (meaning: Let me prove you that you're wrong).