how did aleister crowley know the correspondences in 777?

The man was the spiritual equivalent of P.T. Barnum…

Aleister Crowley travelled a lot and learned many religions and cultures. He learned the correspondences by comparing the various systems to each other, but he focused mainly on the Egyptian mythologies. A lot of what he learned was simply following the work of those before him, but he certainly was very intelligent in his own right and worked hard to understand it all better.
The biggest problem with Crowley was his massive ego. Most of what he wrote is unintelligible unless you have a very thorough understanding of Egyptian symbolism, as well as how it relates to astrology, tarot, Hebrew, etc. In my opinion, he wasn’t always terribly clear about what he was trying to say, but of all his writings, 777 seems to be the most useful and straight-forward.
If you truly understand the use of the correspondences, everything else will eventually fall into place.

DID YOU MOTHRA YOUR UNDERWEAR YET?

Some he learned from people in certain circles.
Others he brought in from his own studies.
He then sought to place all the data he had amassed into correspondence tables.
He said that it was everyone’s job to keep working on the correspondences.
That means they are not set in stone, and that anybody can rearrange archetypal correspondences in any way they see fit (i.e. the Great Heh-Tzaddi Swap).
But don’t necessarily listen to what Crowley said (or me for that matter), because every man and woman is a star!