Here's the thing. First of all, let me start off by saying that I know exactly the type of person you mean, and yes, a lot of runners are like that. Let me also say that I try like hell not to be that person. I don't talk about my running unless its to other runners, and then I try not to mention times or training or anything specific. But, to answer your question, running attracts a certain type of person. They tend to be addictive personalities, be slightly obsessive-compulsive, and be very, very dedicated and immersed in what they do. So for a lot of runner, running becomes more than just a hobby or a way to work out, it becomes a way of life. It consumes them; every waking second is dedicated to running. They become one-dimensional in their dedication.
Add to that the fact that running (and endurance sports in general) is an activity where you can see massive improvements over a period of time, and they get excited about how much they have improved. Sometimes they may even get so much better so quickly that they impress themselves and want to share. Unfortunately, the addictive personality and one-dimensionality rear their heads then and they become overwhelming and tend to brag, sometimes deliberately, sometimes not.
Also, to become a good runner, it takes a certain cockiness and arrogance, along with a strong dose of desire and hunger. These very alpha male characteristics tend to be bundled with bravado and a desire to brag. You can't be good at anything if you're meek and complacent. You have to want something bad enough, have a competitive enough spirit to endure the pain and sacrifice and want to win above all else. If you're good enough (eg- elite), chances are you don't have anything to prove, and sure enough, of all of the fastest runners that I have met, had I not known they were elite, I never would have guessed. They have all been very unassuming and didn't need to brag about anything. But for those who are not quite elite (age groupers, D3 runners, etc), they know their place in the grand scheme of things and still feel the need to prove to themselves and others that the sacrifices they make, the work they put in, is worth it. And so a lot of sub-elite but still very good runners, or decent runners, feel the need to constantly brag to prove that its worth it.
By the way, speed has nothing to do with being a runner versus a jogger. Runners run because they enjoy it. Joggers run because they enjoy the benefits. And Runners (capital R) run to push themselves to the limit.
