Looking at the instructions, it appears as though it was designed to mount to a brick wall.
You could try installing it with two screws on one side into a wall stud, and the other two into the drywall using something like this (Project Source 18-Pack 1-3/4-in L x 1/8-in dia Standard Drywall Anchor (Screws Included) in the Anchors department at Lowes.com) , but I'm not guaranteeing that will work.