In the physical world this has a very literal meaning. The useful work done is the energy applied, which is force times distance. Therefore a small force applied over a long distance is the same amount of work as a large force applied over a small distance. The trick is converting the one into the other.
The simplest device for creating leverage is the lever. A lever is a stick which rests on a fulcrum near one end. When you push the long end of the stick down a long ways, the short end moves a small distance up with great force. With this device a man can easily lift several times his own weight.
But leverage is also used with more metaphorical meanings of the word force. For instance, in finance people think of money as "force." Given a relatively small initial amount of force (money), you can use that as collateral for a much larger loan, which gives you a larger amount of money (force) to throw around. Therefore in finance it is common to call debt "leverage."