It's a very vague phrase when it's not referring to the ball itself. (When it refers to the ball then it's normally talking about getting the ball away from the goal area, as I'm sure you know.) In the first one I'm pretty sure you're right, it does mean "begin play": I think they have probably kicked off. One possible alternative is meaning that they have made the first score - it's not the game but the scoring which has started - but as I say I think your understanding is probably the right one here. The second clip is too short (sorry) for me to be sure, but I suspect it also does mean the same thing.