The baseball rule book does not get additions very often. On top of that, casual fans cannot recite very many rule numbers. There is one circumstance where these two rarities become a reality: Rule 7.13 (old format).
Some folks call it the Buster Posey rule since his injury was the catalyst leading to the rule. It is the home plate collision rule.
Here it is in all of its glory:
7.13 COLLISIONS AT HOME PLATE.
(1) A runner attempting to score may not deviate from his direct pathway to the plate in order to initiate contact with the catcher (or other player covering home plate). If, in the judgment of the umpire, a runner attempting to score initiates contact with the catcher (or other player covering home plate) in such a manner, the umpire shall declare the runner out (even if the player covering home plate loses possession of the ball). In such circumstances, the umpire shall call the ball dead, and all other base runners shall return to the last base touched at the time of the collision.(2) Unless the catcher is in possession of the ball, the catcher cannot block the pathway of the runner as he is attempting to score. If, in the judgment of the umpire the catcher without possession of the ball blocks the pathway of the runner, the umpire shall call or signal the runner safe. Notwithstanding the above, it shall not be considered a violation of this Rule 7.13 if the catcher blocks the pathway of the runner in order to field a throw, and the umpire determines that the catcher could not have fielded the ball without blocking the pathway of the runner and that contact with the runner was unavoidable.
Without reading the rule most fans believed this rule outlawed collisions – like in high school. A closer reading of the rule shows this is not the case.
What does the rule say? In a nutshell, part 1 says a runner cannot go out of his way to crash a catcher. If a catcher is set up away from the plate, a guy cannot get a cheap shot on him. The runner has to make an attempt to get to the plate.
Part 2 says the catcher cannot block the plate without possession of the ball or without fielding a throw that takes him into the baseline.
What happened in this play?

The catcher had the ball before the runner got to the plate. Once he has the ball he can block the plate. Also, the runner did not deviate from his path to the plate. He can (and actually does) touch the plate.
Since neither provision of the rule are violated this is a…wait for it…a completely legal collision. It is a baseball play. The umpires correctly ruled this is an out and the out stood after replay.
Home Plate collisions are covered on page 59 of RuleGraphics.