I am new to baseball, really love it but for me as European the rules are bit unclear. I am confused by this video of Javy Báez. From what I understand as Javy is running back to home plate another player is running there. My question (1) is why the catcher was trying to tag the runner? Maybe so he cannot run back to 3rd base? But anyway catcher was already standing on the plate and had the ball in his glove before runner touched the plate, so I would say runners only option is to run back to 3rd base but (2) as he touches home plate referee signals that the runner is "safe". Why? Many thanks.
There's a bunch of stuff going on here. The most important distinction is between runners who are "forced" and those who are optionally running - a runner is "forced" if they are no longer entitled to the previous base. This always applies to a batter who has just hit the ball (a "batter-runner"), but also applies to runners who are required to move because the batter-runner is coming up behind them - e.g. a runner on first base when a hit is made, a runner on second when a hit is made if there is also a runner on first. A runner on second when a hit is made is not forced if there is not a runner on first as they can happily just stay on second.
A runner can always be put out by being tagged; a runner who is forced can also be put out but a fielder being in possession of the ball while touching the base they are running to.
That hopefully explains why the catcher was trying to tag the runner from third - the runner was not forced as they started on second with no runner on first.
What this doesn't explain is why the first baseman didn't simply step on first base to put Báez out and then throw home to prevent the runner advancing. There's no good answer to that - the first baseman just totally and utterly screwed up, apparently forgetting the rules of the game they play professionally.