"You don't get to say that" means just what it says -- Mitch is saying that Matt Brody is not allowed to be offended by the expression "You people"
In the U.S. people can be very sensitive about racial issues and anything that even seems like a racist statement. The phrase "you people" is sometimes used to refer to people of another ethnicity, for example:
Why are you people always [doing X]?
It's a racist statement because it stereotypes the behavior of everyone of that ethnicity. It also suggests that the person saying it would have chosen a more "colorful" racial epithet than "you people" if it was socially acceptable, but restrain themselves when it might get them in trouble.
This is more about culture than language. Here Matt (Zack Efron, who is white) takes offense to the "you people" comment made by the black police officer Ellerbee (who is black). Ellerbee is not being a racist, because by "you people" he means "you lifeguards". Mitch (played by Dwayne Johnson, who is mixed black and Samoan) reprimands Matt about being offended, since he is not actually a racial minority. He's not any ethnic skin color -- he's "just tan".
The joke is a play on expectations. In most contexts, "you people" is only racist when spoken by a white person. Here's it's a black police officer saying it, and a white person is offended (because he's "tan").
Even this light humor might be offensive rather than funny, but in the context of this otherwise silly movie it's fine, you really barely notice it. It's a throwaway joke in a movie full of much better jokes, although even those are only really funny if you know something about the TV show the movie is based on.