Bluetooth is a wireless communicaitons system which, compared to alternatives, has the shortest range, is the slowest, but uses the least power. It is used to communicate with devices which are usually close and don't need to send or receive a lot of data quickly, like wireless keyboards and mice, and earbuds. I find this works reliably inside up to about 10 meters, although various versions of Bluetooth (you didn't think there'd be only one, did you?) are designed to theoretically work from 1 up to 100 meters.
Earbuds are very small, so their battery is limited, they are generally close to the device they are communicating with, and the amount of data they send is relatively low, making Bluetooth an ideal solution. A similar argument applies to keyboards and mice.
It's usually safe to leave Bluetooth switched on, because it uses very little power, and it is not a common security problem, partly because of its short range. Note that, if you switch it off, don't expect to be able to use your earbuds, wireless keyboard, and some other devices any longer.
Bluetooth requires devices to be "paired", which tells the phone (for example) which earbud (or other device) to send the signal to (so you don't end up sending your music to your friend's earbud who might be sitting next to you). With Apple AirPods, it's just a matter of opening the case and confirming the connection, but with some other devices, from not so user-friendly companies, it can be more complicated. You should only need to do this once, though.
By the way, if you were wondering about the origin of the name, it is the name of a Scandinavian tenth-century king, Harald Bluetooth, who united various tribes into one kingdom. One of the main reasons for creating Bluetooth was to merge all the earlier incompatible methods of communicating into one.
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