Re: Why 100 pushups?
I would say that it has both a physical and mental contribution.
On the physical side, you'll gain at least some muscular endurance, which is useful both in sports and everyday life.
On the mental side, you can do 100 push-ups. That's a sizeable amount of push-ups. And the fact that you've worked up to accomplished that goal may give motivation to tackle other goals.
Just my thoughts. I've thought about doing this program just to say I can do 100 push-ups (which I don't think I could do currently), so I see it as a good addition to whatever you do, be it CrossFit, running, or whatever.