Q:

MLB The Show Players League

Some clip in a MLB The Show Players League video looks like some kind of video games.  The guy pointed out by the red arrow seems to be manipulating a playstation controller. 

Is "MLB The Show Players League" some kind of video game? 

A:

Is "MLB The Show Players League" some kind of video game? 

"MLB: The Show" is a video game. 
"MLB The Show Players League" is a league of players, one from each MLB team, playing MLB: The Show, a video game.  
This article sums up "MLB The Show Players League" quite succinctly:

The “MLB The Show” players league will provide fans an opportunity to
  watch their favorite players play the video game [MLB: The Show], while also allowing
  them to interact with them through various streaming services.