Why does my dishwasher back up into the sink, but running the disposal allows it to drain?
There is a very simple answer to your question.  Your dishwasher is draining into the side drain adapter on your garbage disposer. this was a common practice for many years. All the water draining from your dishwasher is being pumped into the upper basin side of the disposer. Solids from the dishwasher are blocking the drain in the disposer, so when you turn on the disposer, it grinds the waste and drains the water. This is actually the way it is suppose to work. Current codes require a separate trap for the dishwasher and not plumbed into the disposer. If you look at the side of your disposer, you will see the hose from the washer attached there.  Only fix is to install a new separate trap and drain for the washer.  Good luck.