My understanding is LiveATC has a network of volunteers with receivers tuned into frequencies used for aviation. That is why you see such a varied mix of airports and facilities. Depending on where the antennas are placed, some traffic comes in loud and clear while other may be more garbled.

The comms are then streamed up to LiveATC's servers. Depending on the reliability of the volunteer's connection or whether their computer is being used for something else, the streams may cut out or be unavailable. Generally the volunteers are very reliable and committed so stream dropouts are few and far between for the airports I listen to. For the most part, everything is very close to realtime.

Each stream is archived in 30 minute blocks. It's a great tool for those going after their private pilots license. Going back and listening to how you work with ATC is valuable as a learning tool. For fun, check out the interesting comms section. You hear lots of classics like "Bad Day at JFK Ground" and my favorite where JFK Ground asked a Comair RJ what it was like to be positioned behind an A380. The response came in, "It's biiiiig" followed by "Lots of wheels"