I understand that you have come to realize it's not right but to put it in perspective...how would you like to go to a site and see your pictures being streamed on someone elses site without your permission? The copyright laws are the same thing for musicians, writers, photographers, artists, sculptors, etc.
To answer your question how radio stations get by with it, is that they purchase licenses from BMG and other licensing agents to play certain music. These licenses are not cheap and therefore the main reason for commercials.
As for music that you purchase, you may record it all you want to play on any device that you own to be played for your listening pleasure. However you may not give or sell that song to anyone unless you give it to them permanently as a gift or sell it to them and destroy all of your copies.
If you own a copy of a song, and you want to share it online with your friends, you may do so, but you can't allow them to download it. This of course is a real gray area and has been disputed many times, so you need to determine how far you're willing to push it.
Mind you, I'm not a lawyer, and these are only things I've heard and researched as a songwriter/musician, photographer, poet and artist over the years.
Hope this helps a little.