"As for the first question: Why are we fighting and opposing you? The answer is very simple:

Because you attacked us and continue to attack us."

And proceeds to list all the ways the US are militarily attacking and oppressing Muslims in the Middle East. It's a long list.

Homosexuality is mentioned only once in the letter, in the next section, where he criticises American society and morals in general and calls it to embrace Islam. This is explicitly an exhortation and not part of the reasons for the attacks (so probably intended as a diagnosis of the symptoms of a moral disease and the proposal of a cure - note that I'm not endorsing it, just explaining its function in the letter).