There are a ton of possible great answers to this question including the desire to pass on knowledge that they've acquired through years of study and personal participation in the field they wish to coach in and on an even more basic level most coaches know that they have a tremendous amount of influence on those they coach and as we often see this can be potentially a really good thing or sometimes a very frustrating experience - see youth sports for all the fun some coaches have trying to keep concerned parents from tackling referees on the field for a possible mistaken call!
Ultimately most coaches have a strong desire to change the lives of those they coach for the better. Whether teaching a diabetic how to eat better for a healthier life or helping and athlete better understand nutrient timing for optimal recovery great coaches go beyond cookie cutter programming to get to the core of how to make these changes. That might include improving specific skills in a sport but even more importantly the best coaches are usually focused on - especially in a sports coaching setting - helping their players become better human beings while they're becoming better at a particular sport. If you're a coach and what you care about most is how many wins you have in a season or how many times you are mentioned in the local press then I'm not sure you really understand what coaching is all about.
For those coaches who have been in the trenches for a while almost all would tell you that there's something especially rewarding about running into someone you helped coach in the past and hearing them yell out, "hey coach" and hearing them talk about the impact that you had on their lives. Coaching by it's very nature is a position of great influence and responsibility so taking a little time to exam your motives for coaching is always worth the time.