The concept these sites use to justify what they do is that of file sharing between individual members. File sharing is probably legal, but the file sharing of copyright material may not be legal.
The MP3 format has changed the face of music. Music and speech can be converted to MP3 files easily. Some detail is lost but this is not detectable on most MP3 players. MP3 files take up much less space on a hard drive than a conventional audio file. This allows personal MP3 players to hold hundreds of MP3 files on their hard drives, whereas they would only hold 10 -20 conventional audio files.
Music download sites hold a database of files belonging to their members and you access members' own computers to download the music MP3 files you are searching for. Many of the music files have viruses in them, sometimes deliberately put there. Many more files contain spy-ware, giving the provider access to data on your Internet use or even files on your computer.
Some of the download sites themselves install spy-ware on your computer and then sell the information gathered to third party users.
Better, probably, to pay the small monthly download fee from paying sites and download your MP3 music legally and in safety. This way the artist gets rewarded for their effort, and it only seems fair to pay them for the entertainment they give us. If you like an artist or group then encourage them to produce more music of the same type by paying the small fee that is asked, and stay legal at the same time.