To learn Tai Chi you need to work with the force of someone else. The empty hand forms will teach you the movements but push hands practice allows you to apply them.
We also have two person exercises that focus on different skills in Tai Chi: redirection, sticking, and using the force of the whole body.
The transition of weight between the legs and keeping the center of the body stable is an important part of fighting. In Tai Chi when doing push hands you're trying to sense the center of the other person while concealing your own. If you're aware of where their center is and when it changes to a weak position, then you can overcome the opponent. Anytime you have to move, you may shift into a weaker stance and if you're opponent is aware of it they can use that to push your center off balance.
Trying to push a partner off balance teaches you to feel how every part affects the whole. The idea is to make your body work together as one integrated unit. Think of it like a group of people tied together. If everyone tries to go in their own direction, no one will be able to get anywhere efficiently. Once everyone picks a direction, they have to learn to move so that they don't step on each other's toes. Communication on what to do as well as practice at doing the actual stepping is required. With enough time and energy the group becomes efficient and moves as one.