Well, I came to realize that one of the reasons they are so effective at climbing, running fast, jumping, is because their muscles are all balanced. Think of a guy at the gym who is only obsessed with his top half and neglects his legs. He may have a superb figure up top, but if he tried to run fast or jump, he would fail miserably. With cats, as they are balanced all over, and also very flexible, it makes them superb climbers and sprinters. Lets look at the sprinting part - My eldest cat is extremely fast, I honestly haven't seen a cast as fast as her, and she is also very strong. Compare her to Usain Bolt (the worlds fastest man). He is extremely powerful, explosive, flexible, but his muscles are all balanced. He doesn't just have strong legs, or just a strong upper body, all of his muscles are strong and balanced by their opposites.
Being flexible, powerful, explosive, strong, are all attributes of being a true athlete, and if you want to have a body like an athlete, then you have to train like one. By having your body in a 'balanced state', you will feel at your best.
It was strange noticing that whilst watching my cats, but they are incredible animals. They stretch regularly, which is a huge benefit to their explosive power. Stretching is under rated in the fitness world, but it keeps your muscles supple and ready for action - as with cats.
The main lesson here is to train your body to be balanced - don't neglect certain areas of your body, as this could lead to injury, aches, pains, soreness, and will also look odd! Muscles in your body need to be supported by others, so train them equally.