Why do gay people have a funny accent?

That's a good question, but if you want an exact answer you may want to ask a gay person.
My thought is, they want to be noticed and stand out.

If you are gay, only about 11% of the gender you are attracted to are also gay. This immediately limits your options. If you don't know whether someone else is gay or not and are afraid to ask, this also limits your options. What you need, then, is a way of identifying people's sexuality.

The most obvious would be congregating in a place intended for a specific sexuality (i.e. a gay bar), but there are other ways. In my country, certain things like having a visible handkerchief in the back pocket of your jeans, was a symbol. There is also gay slang that straight people aren't party to. Finally, you could publicise that you are gay by acting a certain way. I'm not suggesting this is the reason but it's a possible reason.