The way it works is that a number of enthusiast have real-time gadgets which receive radar-type transmissions from aircraft. Many of these include details of callsign, altitude, speed, etc. They then feed this stuff via the internet to the various sites, such as FlightRadar24, which then streams it on the internet. Well, that's roughly what happens unless you want to get technical. Many aircraft will not appear for various reasons.
ATC have nothing to do with this as they have their own specialised radar systems.