This bill overrides local ordinances that may provide protections under local Public Accommodation laws. What is incorrect in that Statement.
You are correct though, in most of Arizona it is already legal to discriminate against gays and lesbians, but this law does is expands who can be discriminated against. Claim a serious religous belief and this law protects you even if you are discriminating against blacks, asians, Jews, Muslims, etc.
A few years ago there were cases of Muslim cab drivers in another state who refused to carry passengers carrying alcohol (not drinking, carrying closed containers) and to carry service dogs for disabled people. They were found to be in violation of the law. Under this law they would be exempt from Public Accommodation because their sincerely held religious beliefs (as defined under Shaira Law) would exempt them.