While the original Internet machines weren't Unix machines, the ones that allowed it to scale up were. Unix is a pretty "optimal" platform for creating and deploying Internet applications as a result.
Programs can be written to do a single thing, spawning additional work processes as needed. And they can, and by default do paste together subsystems.
Providing an easy-to-program way to talk to processes over a network
Microsoft would have us think that they invented "WinSock;" it was actually the nice BSD people. Microsoft probably wasn't even first in coining the term "WinSock"...
Filters using STDIN, STDOUT
Providing a base "protocol" whereby programs can be built to generically talk to anything
This is a true fundamental of The Unix Philosophy
On most other platforms, linking programs to communicate together requires that you write a program. Under Linux, that "program" can be exceedingly simple. For instance, you can connect 3 filter programs together using: % cat input_file | filter1 | filter2 | filter3 > output_file
Allowing rapid prototyping to "paste" together communications flows
Several Usenet flame wars have taken place over the issue of whether or not Free Agent (a popular "free" Windows newsreader) is better than Linux news readers. Brian Mueller claimed that FA is better because it "uses one program to do what takes 3 under Linux." (It combines an NNTP news client, NNTP news server, and text editor.) Similar arguments could be made for mail clients like Eudora.
Under Linux, the separation of those roles allows us to insert additional functionality that Windows users are restricted from even imagining.
I can use outside programs to redirect mail in as complex ways as the mind can imagine.
I can use Web Proxy servers to improve performance, improve privacy, and eliminate unwanted information. And share many of these services to Wintel users.
If your office or school needs to have an Internet connection, have that connection go thru a Linux box. You can put in proxy services (perhaps via use of multiple proxy servers) to:
Cache requests, so that when people visit common pages, the data doesn't need to travel over the connection additional times;
Centrally log accesses, so Big Brother can watch;
Block access to "problem sites;"
Block (potentially in both directions) transfer of sensitive information.