Hold the cabinet in place against the wall in the desired final location. Place the level on the top of the cabinet and adjust the cabinet until it's level. Have an assistant hold the cabinet firmly against the wall.
Drill three pilot holes through the sturdy part on the inside on the top of the cabinet through the wall. Insert a 3/16-inch drill bit rated for wood into the drill and secure the drill bit. Make sure the drill is straight as you drill the pilot hole. Drill the pilot holes at least 2 1/2 inches deep.
Remove the cabinet from the wall.
Install the molly bolt anchors into the pilot holes in the wall. Remove the screws from the molly bolt anchor with the drill. Push in the anchor of the molly bolt into the pilot hole in the wall. Gently insert the anchor completely into the wall with the hammer so it is flush with the wall.
Hold the cabinet against the wall again. Have your assistant hold the cabinet while you secure it in place. Line up the pilot holes inside the cabinet with the anchors on the wall. Secure the cabinet by drilling the screws through the pilot holes in the cabinet into the anchors. Drill all the screws before letting go of the cabinet.