You have to get over the Fear completing Fear bubbles. These appear in the pointy bubble on the right.
So for Fear of Fire you will have to wait for a bubble related to that Fear, and then complete the action listed. I believe you have to extinguish a fire, but it doesn't count unless you have the bubble.
Make sure you read the instructions in the Fear bubbles properly because you might miss a detail like having to be in a specific mood or doing 2 specific actions in a row.