IANAL, but here's my reading:

The summary of the actual bill reads: "Prohibits a governmental entity from substantially burdening a person's exercise of religion, even if the burden results from a rule of general applicability, unless the governmental entity can demonstrate that the burden: (1) is in furtherance of a compelling governmental interest; and (2) is the least restrictive means of furthering the compelling governmental interest.[...]"

Religious groups are already protected from specifically targeted laws by the first amendment -- for example "It is illegal to worship Allah" would clearly be unconstitutional. However, they are not by default exempt from 'generally applicable' laws, before RFRA. For example, the law banning the use of peyote could still override the religious freedom of practitioners of Native American religions that require peyote use in ceremonies. So RFRA attempts to extend the exemption a little bit, to say that religious groups are exempt "unless it's for a really good reason, and the law can't be relaxed and still further that cause". This provides a concerning amount of wiggle room -- though calling it a 'gay discrimination' bill is a bit of a leap IMO. Here's how it can be used that way, though, with precedent:

The Hobby Lobby decision people mentioned (I for one, completely missed it at the time) was essentially that a Christian-owned company was not obligated to provide Plan B and other contraceptives in its healthcare plan because it conflicted with their religious beliefs, but only because there were less restrictive ways of furthering the 'compelling government interest' -- that is, the goal of the contraceptive mandate was to ensure access to contraceptives by all women, but there were ways of ensuring this that didn't directly burden Christian-owned companies (such as being provided directly by the government). Notably, this also means that the Hobby Lobby corporation was 'a person' in a sense, as it related to RFRA. So I believe what people are worried about is other Christian-owned stores claiming exemption from the obligation to provide equal service to everyone because it went against their religious beliefs; the question would be whether the intent behind nondiscriminatory service laws could be upheld equally well by laws that didn't require Christians to serve breakfast to gays. Which seems like it would go to court almost instantly about (probably?) get struck down.