Linux development was started by Linus Torvalds when he wanted an affordable way to use a Unix-like operating system on his home computer. He used the GNU tools by Richard Stallman et al to provide his operating system kernel with userspace tools.
People liked this idea of a free (both as in beer and as in speech) operating system and lots of people joined in in the development.
Companies like Red Hat decided that providing support for Linux would be a good business model and that to do that they should pay people to develop it.
Some companies decided this operating system was just what they were looking for for their cable set-top-box or other embeded system and, like Red Hat, some paid people to work on Linux development so that their needs could be met.
Google decided they wanted to take on Apple in the mobile phone market and decided that Linux was the perfect base for an operating system to compete with iOS so paid a bunch of developers to make Android.
In the mean time lots of businesses and institutions decided to deploy Linux on servers to run websites and on computational clusters to simulate everything from car crashes to the big bang.
Lots of home users loved this OS they could do what they wanted with without licensing agreements or a committee of marketing folks telling them how their desktop should look.
Lots of programmers liked being able to see the source code for their OS.
So, Linux is important to different people for different reasons but it is primarily important because it is free -- as in anyone can use the OS as they wish and can see the source code and modify it as they wish.
Why Linux succeeded in the words of its creator:
http://www.bbc.co.uk/news/technology-18419231
