Mountain biking is what lead me to getting a dirtbike. My knees don't like pedaling though so I've had to really cut back and the dirtibke is a natural substitute and a little more fun.
There is no doubt that riding a bicycle regularly can get you into great shape, but so can a lot of other things that are often cheaper and more convenient. Cycling has also caused me some moderate long term knee pain after several years of intense riding.
Skill wise it depends a lot on how you mountain bike. I think the biggest benefit is not particular skills (ie crossing logs) but the idea that you can shift your weight to control the machine, that you can huck a 3 foot drop on the trail, or that you can unweight the bike and float over some obstacles. Riding aggressive AM/DH/FR changes your idea of what is possible on a bike and I think that carries over really well to a dirtbike even though a lot of the skills and movements and possibilities are different.
I remember hucking a moderate drop for the first time on my two stroke. I weighted the front as I approached the edge and then leaned back as the front tire left the ground just like on the mountain bike. With a little twist of the throttle the bike launched me forward like a rocket. I added a whole new dimension. Can't do that on a peddle bike.