Does being fit or muscular help to stay in better shape as people age? I have to ask as intuition would say yes but there don't seem to be serious studies on the subject. Also, some athletes are said to end up with a bit of wear and tear early in life. There could be a point beyond which exercise and build could be detrimental.
There are several aspects to aging, and depending on what you do will affect how you age. It's not as simple as "just be fit and you'll stay healthier". Apparently there is a book on the subject that will provide a much rounder picture than I can provide here.
Outside of these major quality of life and health concerns, proper diet and hydration affects your skin tone, fat content, blood sugar levels, etc. The sad truth is that there are no guarantees that will give you a high quality of life until you die. However, there are things you can do to increase your odds.
Depends on what you mean by "aging better". Aging has a lot of different aspects, for example the loss of health (e.g. increased cardiovascular risk), loss of physical fitness, and deterioration of physical appearance.
There is no question at all that exercise prevents age-related health problems, and there are countless studies on that subject. Here's a study showing that exercise reduces cardiovascular risk. The same is true for physical fitness: Obviously exercise helps you stay fit as you age, since that's the main purpose of exercise in the first place.
Perhaps you are more interested in the age-related effects on appearance, i.e. does exercise help us look young as we age? This is a more complex topic, since appearance is fairly hard to measure.
One part of it is wrinkles, and I would argue that staying muscular is definitely a great way to prevent body wrinkles, by keeping the skin tight. In addition, skin hydration is a big factor in preventing wrinkles and exercise keeps you and your skin hydrated by strengthening your heart and improving your blood circulation.
Another part is how "healthy" you appear in terms of your skin tone, which is also affected by cardiovascular strength (blood circulation) and as such definitely associated with exercise.
Yet other parts are your stress-resistance, energy levels and emotional stability (sort of your "behavioral appearance"), all of which may decrease with age, and all of which are significantly improved with physical exercise.
So there's no doubt that exercise slows almost every aspect of aging. Is exercise also associated with increased risk of injury? Yes, but injury is something you can control and prevent to some extent if you're smart, while the creeping effects of age-related decline are not (unless you exercise).
The "wear and tear" kind of injury is mostly associated with excessive exercise, such as running the length of one or two marathons every week of your life. It's entirely preventable by moving from low- or moderate-intensity exercise to high-intensity exercise of shorter duration. But that's a different topic.