Enzymes work by decreasing the amount of activation energy needed to start the chemical reaction.
Basically, enzymes work by attaching the substrate (the substance that wants to be catalyzed) onto the active site of the enzyme. Then, the enzyme breaks some of the bonds within the substrate, making it have weaker bonds.
These weaker bonds are easier to be acted with, and upon, thus requiring less energy (ATP). It is through enzymes that lots of cellular activity can be done quickly, efficiently, and at a low ATP cost.
Here's a good graphic showing the process...