Edit Coming Back to This Question After A Few Years
Having continued to use more and more boost libraries, I thought I'd update this question to give a solid case for why you should use boost when the description of the product matches your desired functionality. This will convince even nay-sayers. Download openSSL, try to make a client and server application with it. Now try and make that work on every platform. Then, download and use boost::asio::ssl to make the same application. If you're not convinced that boost is the right place to look for clean, well optimized, peer reviewed, cross-platform code, this simple exercise will convert you.
You don't see a ton of indie or small to mid size development firms using boost because it's a massive and powerful wild beast that isn't easy to tame and you're basically on your own when trying to learn how to use it. The documentation is lacking in a few ways (see long version) and "the community" around the project either seems to be missing, scattered or inactive (compared to other projects).
Very Long Winded Version:
I realize there is already an accepted answer but as someone who actually uses boost in nearly every project I do, I thought I'd post an answer.
I remember when I first got poking around in boost and honestly I had no frigging idea what was happening. Boost is not very well documented at all. People might disagree with me on that I'm sure because there are tons of snippets of example code and a comments and such, but it is all very cold and vague as well as difficult to navigate.
Also it seems difficult to find any place where you feel like you've found "the community" around the project. In fact the community seems non-existent, or nomadic. Unfortunately even their mailing list has been trolled by so many leech sites that you can go down this rabbit hole always looping back around to where you started.
These two factors make learning boost a rather daunting task. Even if the technicalities of using boost aren't excessively complex, it is a massive set of libraries and staring it down when all you're armed with is a few code snippets and scattered pieces of the mailing list from the darkest corners of the internet... well you get the idea.
I got into tinkering with boost around version 1.45 and it's only now in version 1.52/1.53 that I feel comfortable enough to use it in production. There are so many things to get used to and remember, even simple things like how you've configured boost and remembering that configuration, because how the libraries are built and function can vary wildly based on your preferences at compile time due to how customizable things are.
However, make no mistake, once you can wield boost, you've gained a powerful weapon for rapidly building solid, cross-platform programs. Just take
boost::asio for example. You can write an immensely powerful, scalable and rock solid cross platform asynchronous web server in just a couple hundred lines. I've written multiple clients, servers, proxies etc over the years with just a few hundred lines of code each that have yet to fail me yet, and can port them from platform to platform in minutes.
As others have pointed out, larger companies are usually stuck with legacy stuff or like to roll their own which I completely understand. There is also this really silly thing I've heard of and encountered where dev leads and or project managers forbid using boost because it's "too big". This is a nonsense opinion held only by the ignorant who probably believe that boost is 1 single shared library (lulz).
As for WHY I choose to use boost
I'd say I use it because as you imply in your question, it's "the" C++ library. Boost is viewed in the C++ world as the swiss army knife of things that eventually you're going to need to use. So the idea is that if there is a need, there should be a highly performant, and portable version of it in boost. Big companies contribute to boost, very educated people with impressive resumes contribute and maintain it, and when a new standard of C++ what being developed, people usually look to boost to see what parts of it should become ISO standardized C++.
So if I need to add some functionality that there is probably an existing library for, the first place I'll look is boost just because I'm pretty safe in betting that it's pretty well optimized, portable, it will be supported and maintained for a very long time and bugs will be found and dealt with. In the open source world those qualities can be very difficult to come by.