There's a myth that to be light on your feet, you have to be in great shape. It's true that being in great shape will allow you to move faster than one who is not. But this is not the same thing as being light on your feet, which is about how quickly you start to move, not how fast you move once you get going.
How quickly you start to move - i.e. how light you are on your feet - is mostly a technique thing. If you watch the top players or anyone who seems to move quickly, watch how they take a slight hop as the opponent is making his shot. It is this slight bounce that prepares a player to move almost instantly. Those who simply stand there, waiting to see if and where they have to move, lose the bounce that comes from this, and so they are very slow to start to move.
There are players a hundred pounds overweight or in their seventies who are light on their feet, and players in tiptop physical condition who are not. This doesn't mean they move fast; it means they get started quickly, and so while they may not cover a lot of ground, they often seem to always be in position - because they are never flat-footed and stuck in place. It means they react to shots very quickly because they are always ready to move.
Are you light on your feet? There's a simple test. When you are caught off guard, such as against a net or edge ball, do you step to the ball, or do you just reach for it? If the latter, you are flat-footed.
To see this little bounce that players do between shots, you can watch just about any video of top players, and focus on one player. You'll see this slight bounce as the opponent is hitting the ball. Few people see this because 1) it happens too fast, and 2) when watching a match, most viewers are watching whoever is hitting the ball rather than the one who is not. Ideally, see it in slow motion. Here's video of Zhang Jike doing multiball. Watch seconds 45-50, and you can see (in the slow motion) the very obvious bounce he does between shots.
Or just watch video of just about any other high-level match, such as the video highlights (4:50) of the Men's Singles Final at the 2015 World Championships. Focus on either player (Ma Long in Black, Fang Bo in orange), and watch their knees. For example, in the very first point, see how Ma Long returns the serve, and then makes two great forehand loops. But it is the slight hop he takes before moving to each of these shots that allows him to get the quick start that positions him for these shots.
Another thing that's important is the foot positioning. To be light on your feet, use a relatively wide stance in a slight crouch, knees pointed slightly outward, with weight on the inside balls of the feet.
The thing to emphasize is that you can be overweight, old, and have bad knees, and you can still take this slight bounce - it's just a matter of making it a habit. How do you make it a habit? Like anything else - practice. But the nice thing is that this is one of those few things you can practice doing just as well in a match as in a drill, so there's no excuse for not practicing it. Just do it.