paying royalties is how you get around buying the rights.
The actual methods may be slightly different, but I think for the most part copyright has the same legal principles regardless of jurisdiction. (A quick google search tells me Sweden is pretty similar to Australian copyright law)
So let's assume you own the copyright in a piece of music (and it gets complicated here, because there's a few different types of copyright in each music piece). You have the right to prevent people from copying your work (e.g. illegal downloading comes into play here). You also have the right to control how your music is broadcast and who gets to listen to it.
The basic principle is that if you have worked hard on creating this great song, you should be able to have your rights protected so you can make some money off that song as a reward. So, in Australia there is an agency called APRA. If you plan to play music to the public (most common example is a retailer) then you pay APRA a fee every year and they give a portion of that fee to the musicians, or the copyright owners. So in this sense, you are "borrowing" the right to use that music. Because you don't actually own that right, does this make sense? If you did own the right, then you would be able to allow other people to play the music.
So if you want to play songs that are on the radio, you would need to pay a licencing fee to do so. If you just want to play music and don't care what it is, then I would look into asking local bands if you can play their stuff. This is the same principle; they are giving you the exclusive right themselves to play the music.
So, the above principles will apply, it will just be a matter as to who the agency is that you pay the licencing fee will be.