You download the scrobbler and keep that always running in the background, then it sends your plays to your last.fm account each time you listen to a song. If you use Spotify, it has scrobbling built in (if you add your last.fm account in the settings). I don't think you can actually listen to music on the site.
After you've successfully downloaded and installed it, open your music player. After that, it should open the lastfm scrobbler app as well. Just log in and start listening whatever song you want to. I hope this helps.