Why does the water from the disposal side of the double sink drain into the dishwasher?

Probably the drain hose from the dishwasher isn't high enough to keep the water from running back . The hose should be above the level of the disposal at some point.

Some states require an "air gap" which is a device that is placed in the drain line to prevent this. Some states require a simple high loop in the drain hose. Many dishwasher manufacturers are now installing backflow preventers in the dishwasher.