What is your centre? I think of the centre of your body as the part of your body from your shoulders to your hips – the part that would make a rectangular box if you drew a line between your shoulders and your hips and across your shoulders and hips; your torso, the part that excludes your head, arms and legs.
Moving from your centre is important for leading. If you want to move your partner, clarity comes from moving your centre, not your extremities. Particularly for leads, if you think about making movements across the floor by moving your centre, you don't think about what your legs are doing. Therefore, your movements are clearer to your partner and they will find it easier to understand the direction you are leading. Your legs move in order to help you get your centre from one place to another, but they don't initiate the movement. If your legs do initiate your movements, that's when you step on toes, because it's your centre moving your partner's centre out of your way that makes the space for you to step into. If you are moving from your centre, your partner can easily determine your movements and you can move easily as one.
In general, initiating movements from your centre will help you get nice lines and move with grace. It's like your movements are an explosion of energy from the centre of you and you feel more in control of them.
Here's an exercise: Try raising your hand into the air and then bring it back down. Now do the movement again, slower. Feel which part of your body initiates the movement. Was it your shoulder, your wrist, your ribcage? No right or wrong answers, just observe. Now see if you can initiate the movement with some other part of your body. Just ask the questions, how do I raise my arm? What happens? and see what answers come.
Whatever answer you get, try to raise your arm again using a different part of your body to initiate the movement. It will feel different, the end result (hand above head) won't change, but the quality of your movement will.
I hope through this exercise you can feel that by initiating the movement from anywhere within that central box (in the picture), the movement of an extremity, such as a hand over the head, will feel more connected to your whole body.
Striking nice lines and having confident and clear movements comes when the movement of legs and arms comes from your centre. It's through the centre of your body that everything (from your feet to your arms – and even your partner) is connected.