Let's start with the basics. The goal of weight training is to strengthen not only your muscles, but the connective tissues that literally hold your entire body together. Your body is only as strong as its weakest links - the joints and connective tissues that can become easily strained with intense exercise. Running, for all of its joys and wonders, delivers a huge physical strain to the body. Weight training strengthens your body from the inside out, making it possible to train harder and recover faster.
Weight training enables the body to burn fat and gain muscle at the same time.
The buzz now is all about body fat levels - and rightly so. When I began to lift weights, I lost a grand total of two pounds in my first three months of training. Now, that ain't much to write home about. I can't remember how many people stopped me to say how great I looked, and ask what I was doing to lose weight so quickly. And my race times were starting to improve as well, without any major changes to my training routine.