If I want somebody to come close to me that's what I say, not "Come at me!"Come close to me.
I concur with pob14 for the most part, especially in that "come at me bro" (including "bro") has become a set phrase and also that learners may have difficulty with the subtleties of when it is and is not appropriate to use, so probably are better off avoiding it unless they have an especial facility with that kind of slang usage.My older daughter calls her female friends “bro” all the time. (Also our hamster, but he is male.) She is not African-American, but she may be low-class; I certainly am.
I think “come at me, bro” has reached the level of a set phrase among the college-age group.
Edit: I should note that I do not recommend this usage (of “bro,” or of “come at me”) to English learners.