It's amazing to notice that live stream sites like doitlive.tv can deliver videos on very low bandwidth (as low as 25kbps) home connection. Could someone explain me the technology behind such sites and how they managed to deliver under such low bandwidth?
I've been working closely with a few major companies at work lately on this very issue. First and foremost, as already mentioned in other answers, a Content Delivery Network is utilized to provide optimum distribution.
A CDN is basically a worldwide cluster of servers that holds many copies of a single resource. So, when you request that resource from, say, New York City, you get the version of it that is PHYSICALLY closest to New York. There are many great explanations of how CDN works.
Your question about bandwidth involves a technology called Adaptive Bit Rate Streaming. Let's say you have a live broadcast streaming out to the web. As it's streaming, there is a piece of technology called a segmenter that chunks up the whole file into small packets that can be pieced together later. Each segment is encoded to various resolutions and capabilities. So, as you, the client, are asking for the files, the CDN can tell how strong the connection is. If it is low, the Adaptive Bit Rate Streaming gives you a lower resolution file. Have you ever noticed how when you start watching video online, it's low quality but improves over time? This is a perfect example of this technology. As your connection buffers further out and is fast, you get higher quality "segments". Hit up the websites of companies like Level 3 and Akamai and you can read a lot of their white papers on how it all works.
Here's an article on Adaptive Bit Rate Streaming.