The song says, "Don't worry. Be happy." But that on its own may not be enough for any specific individual who wants to cut down on worrying.
I actually think the song make a lot of sense. However it is not always possible to not worry and in some cases we should worry. But we have to worry about the right things.
But I think one can "learn" or teach themselves how to worry about only the important things, which to me is basically to start being critical of yourself. So each time you worry about something, stop for a moment and take some time analyzing it, what exactly is so important about this specific thing, that you have to worry about it? A lot of them are really not worth it. For instant being bad at loosing in games might make you get very upset, so obviously you would worry greatly not to do it, but when you think about it, there is most likely two reasons why people don't like to loose in games, they seem worse or less competent than those that won or they might not like, that someone could potentially make fun of them for loosing, which they don't really have a way to defend against, since they lost.
When you know this, you can also quite easy figure out, that loosing a game have nothing to do with how competent you are. And learning yourself to "play along" with someone making a bit of fun, can actually make playing and loosing games fun as well and remove the worry, and teach you to not spend to much energy on it.
I have a small nephew, which when he was about 2.5-3.5 years old, were an extremely bad looser, to the point that he could start crying and get upset, if someone else that he voted for to win were loosing, even though he weren't actually playing himself. Which were pretty annoying for everyone, as you don't really want to play games when you know you have to let him win to avoid drama. His parents kept telling him that its no deal to loose and that its part of playing game etc. Which clearly makes little sense to him as he want to win. So whenever he asked me to play with him, mostly memory games. I told him straight up before we started, in a fun way obviously, that I were in fact the worlds best memory player and that I would not let him win. At that age they believe everything you tell them, but for some reason, it seems to prepare him for actually loosing and that it weren't that big a deal, so rather than getting upset about it, it became a challenge for him, to try to beat the worlds best memory player, and when he finally won, he couldn't wait to tell his parents. So at least now he doesn't care as much about loosing anymore. Since we are talking about a child here, who is not able to really analyze their own thoughts or reasons as adults can, you can help them. But I think basically its the same reason why so many adults tend to worry about to many things that ain't important, because they don't really take the time to figure out whether its something worth worrying about in the first place.
Don't know if what I wrote here makes any sense