The DJ is actually streaming the music from their computer to (probably) a shoutcast site.

When someone is streaming music to a shoutcast site, other people can tune in to that site using a media player (Winamp being one of the most popular, but it's not the only one).

In order for you to do this, you will need to go to Shoutcast then click on the Download/Be a DJ link (or the one I just provided)

While I DJ, someone else set up the station, and thus I have little further knowledge as to the details of setting up/running the station.

In order to DJ, you will also need a broadcaster. I use SAM Broadcaster from Spacial Audio but it's not free, nor inexpensive.

Spacial Audio does have an interface that allows you to use Winamp as a broadcaster, and some people do prefer it, not only because it's free, but also because it doesn't have a lot of settings/controls.