After 10 years of coaching youth sports and observing parents, I think this has something to do with 'mirror neurons', the ability of our brain to watch, learn from and empathize with others. Just like you have an immersive effect with a sci-fi movie and empathize with the characters and feel their plights, so do sports fans. When it's your kid playing of course, it's intensified. Of course, many are just straight up living vicariously through their kids.