Before the "web" there was still the internet (and even some networks before that). Theres a protocol for "news" servers and originally they were briefly about actual news but turned more into discussion groups. Your ISP would have a news server and they would sync this with other news servers around the world. You could then "subscribe" to certain "groups" in your news client and then your news client would download all the new posts made in that group while you were dialled up and then you could read the stuff offline, make a new post in the group to reply etc. Someone started a "group" called alt.bin (alternative category, binaries - ie. files) where stuff would get posted. alt.binz has a z in it because haxxors and pirates like to use X and Z

That's where the app Alt.Binz got it's name from.
Anyway, these news servers still run but don't get used nearly as much, mostly because they got the shit spammed out of them. One of the limitations though is that posts on news servers can only be about 1mb or so, so if you want to share a 20mb file it has to be 20 parts. This is where NZB files come in. They are an index of all the "posts" that contain the parts of the file you want. The NZB's can be shared online legally etc because they don't actually contain any of the files. You'd use this with an app that's more specialised at downloading than reading any news but it still uses the same protocols and the news servers. It would however reassemble the file for you etc.
Now you can use your ISP's news server and use it with "local only" bandwidth but they might now have everything. If you have an uncapped account you may as well use international bandwidth as well, then you can pay for an account at an international news server. These generally will give you higher download speeds and access to more stuff than your ISP's news server.