Coins are used to Level Up your Dragons so they become stronger and can continue the Levels in the Campaign. Your Dragon will need to be at a certain strength to complete some Levels.  
When you click on the Upgrade button of your Dragon you can then click on the yellow Level Up button under the picture of your Dragon to Level Up. Once you have reach the Max Level you will need to Evolve your Dragon with Gems to continue to Level Up.
You will need to Level Up and Evolve your Dragon to be able to progress through the game