It's a question of broadcast rights. Of course nobody can make you stop looking at an event on the day, or delete any footage or photographs you make on your own devices. It's yours, and you can watch it back a million times, and there's practically nothing they can do about it.

However, the moment you broadcast it in some way (for example, upload it to YouTube or social media), you're at the mercy of the owner of the broadcast rights.

Obviously, in the majority of cases, it's in the interest of the rights owner to leave any amateur footage of sports events online. More eyes on the product isn't a bad thing most of the time. It's only when that content causes them a problem (reputation / image-wise, or costing them money) that they'll take action against it.

OT - movies / TV shows / music area bit different, and studios / record labels tend to intervene very quickly, since the thing being uploaded and shared is the product itself, and every unlicensed broadcast effectively costs them money.