In order for a dishwasher to work properly with the soap designed for its use, the water must be quite hot. Most dishwashers also have a heating element in the basin of the dishwasher that will heat the water to the necessary setting. This hot water heats the dishes, and in turn they dry quickly when no more water is being splashed over them.
Dishwashers also have an option for a heated drying cycle. During this cycle, the heating element in the bottom of the dishwasher will heat up the air within the compartment, causing it to dry the dishes quicker than if left at room temperature. This is the same heating element that is used to heat the water for washing. Just make sure that you do not have any plastic items that might have fallen to the bottom of the dishwasher during the wash cycle touching the heating element, or they will melt.
Many times you will see steam escaping from the front of the dishwasher. This is due to the circulation of the electric drying fan forcing the heated, moist air out of the dishwasher. This will run as long as the heating element is on so that the air inside stays hot while the fan helps drop the humidity level, thus drying the dishes.
Of course, the dishes will dry with just air and time as they sit there draining and drip-drying in the racks. This is the most economical option, but there is an increased chance of having water spots left on glass items. The air drying process can be speeded up if you open the dishwasher just after it is finished and the dishes are still hot. This will allow the moist air to leave the dishwasher as the dishes cool and dry.