If you have leverage, you hold the advantage in a situation or the stronger position in a contest, physical or otherwise.
The lever is a tool for getting more work done with less physical force. With the right leverage, you might be able to lift a heavy box. This refers to non-physical situations too: the power to move or influence others is also leverage. Since your boss has the power to fire you, that's a lot of leverage to get you to do what he wants. If your friend owes you a favor, you have leverage to get a favor of your own.