how did the rainbow color become a symbol for gay people?

This has been discussed on here before regarding the rainbow flag. But I can testify that it was not yet necessarily a Gay symbol in 1984, when Jesse Jackson, running for the Democratic presidential nomination, had rainbow campaign buttons and called his supporters the "rainbow coalition". On the other hand, in the late 90s in New York, I was wearing a rainbow yarmulka because of its colourfulness, and some guy accosted me talking, in a hostile tone, about that being an obvious Gay thing.
So, regardless of how far back the symbolism goes, I'm thinking that the rainbow becoming mainly a Gay symbol in the culture at large has to be dated between the mid-80s and the late 90s, at least in the USA.
As to why, I think it makes intuitive sense because of a) being colourful and vibrant and b) representing a whole spectrum -- in this case, of sexuality.

A rainbow is a unity of all the colors in the spectrum of visible light. Sexuality is a spectrum, we would not be here without the whole of it! A rainbow is beautiful, and it reminds us to accept all the population. We're beautiful.

Perhaps it suggests inclusivity and acceptance of a very wide spectrum of people by all and everyone.

Indeed that would be ideal, perhaps not needing that clumsy alphabet soup, for whatever our individual sexual orientations we are all human beings, not items of goods classified in a warehouse database. A rainbow is of very wide spectrum, and it arches over a very wide area.

For that matter, when and by whom was it ordained that the simple word "gay" be turned into a synonym for "homosexual"?

The rainbow is a symbol of hope for all of humanity, irrespective of sexual orientation, gender, race, religion . .

Gay people tend to be more tolerant and inclusive than most, which is perhaps why you made the association.

Cause gays are colorful 🤷‍♀