It's a fairly well known and relatively easy-to-prove rule, and more generally can be stated as:
> The sum of the digits of a base-10 representation of a number n is congruent to n (mod 9).
This has the added effect of being true mod 3, because of course if you have a multiple of 9, or 3 more than a multiple of 9, or 6 more than a multiple of 9, you necessarily have a multiple of 3.
Just google "divisibility by 9" and there will be a bunch of discussions and proofs that show up. I'm not sure if it has any formal name beyond that.
You may be interested in other such divisibility rules, which often get brought up in discussions about this particular rule. Feel free to look up divisibility rules for things like 11 as well.