I have been a trainer and cyclist for 10 years, and I do use resistance training for one very specific reason to improve my cycling: To increase my lactate threshold so my legs can produce more power and I "burn out" less quickly on sprints and climbs.
To do this, cyclists can work in a program of plyometrics and super-slow training. It is *very* intense, and an untrained person should definitely consult a sports and fitness professional to do it correctly.
Like anything you do, your body adapts. By working in explosive plyometrics and intense super-slow resistance training, you can move beyond training plateaus and realize power when riding that you never knew you had!
I have to admit, though, that this stuff is not for the casual cyclist. You have to kind of like torturing yourself in the gym. I'm tall and thin, but I've been training with weights (properly) for 17 years, so my body loves to be challenged. My workout partner, who is a former college athlete and 10 years younger than I, also enjoys our ultra-challenging workouts--but again, her body is also "primed" because of her history as an athlete.
If you have not trained seriously with weights before, you might not enjoy this type of training. But if you are competitive on a bike, and you consult a well-trained fitness pro for help, you will *love* what this stuff does for your cycling production!
To each her own, but that's my two-cents.
There's only one thing better than spending a day on my bike: Spending a day on my bike in good company.