I may be wrong but I think they are referring to the Republican party's position on trans athletes in sports (literally, who is allowed on a girls sports team) being the only thing that matters to some voters.

Its a joke where all Republicans care about is winning the culture wars. Meaning there are boys on the girls sports teams who identify as girls. Its to get the base mad and get them to vote (supposedly).

I think they're referring to the obsession that some people have to trans issues.