While services like Popcorn Time have been simplifying BitTorrent, a peer-to-peer file sharing protocol, for end users, it's becoming easier and easier to look at torrenting as an abstract streaming service rather than a tangle of users passing files between each other.
The BitTorrent team has put together some visualizations that'll help you imagine what's happening here. The team is putting together a browser called Project Maelstrom, which can visualize torrent networks using data pulled from their popular µTorrent client and also promises to allow websites to leverage BitTorrent technology in users' actual browsers.
The team uses a bit of an odd analogy to explain torrenting actually works: you're trying to find a person (the torrent files) at the party and you're asking a bunch of partygoers (peers) where that person is. Those people point you in the right direction, leading you to the person, or people you're looking for. Instead of asking one person at a time, your torrent client can ask 50 peers every millisecond, and what you're looking for is actually some number of peers holding the pieces of that torrent you're looking for.
Of course, torrents are a little more complicated once you take it out of that party analogy. That party doesn't mention "trackers," or servers that handle requests from you, the "leecher." The tracker acts like a matchmaking service, bringing you a list of peers with the right goods. Consider it the RSVP. When you torrent something, you use either a .torrent file or a magnet link to access one or more of those trackers that'll point you to a network of peers that share the files.
When you've downloaded the file, you become can become a "seeder," uploading the files you've downloaded for another set of leechers. Magnet links serve the same purpose, but don't rely on a .torrent file or a tracker—the list of peers is instead generated from a seeder/leecher.
That's still pretty much the short of it; BitTorrent's workings are still somewhat mystifying to the average user, but while interest in it expands, there will always be a subset of users who will want to know how this all really works.