What does the cowboy slang his mean?

"His" in cowboy slang is a shortened form of "his'n" and means "his." It's used as a possessive pronoun, just like "his" is in standard English.

Here's how it might be used:

* "That there's his'n horse." (That's his horse.)

* "I found his'n hat over by the fence." (I found his hat over by the fence.)

It's a bit of a regionalism, prevalent in the American Southwest and parts of the West, and gives a more folksy feel to the speech.