Find the center of the cabinet sides and mark a point on each side of the cabinets at the height you want the clothes rod to hang. If you want to hang mainly shirts or skirts, then around 45 inches off the floor is fine. Full-length clothes need to be around 65 inches off the floor.
Hold a level between the two points to make sure they are at the same height. Adjust as necessary so that the two points are evenly placed and the level is straight when held between them.
Place a wall-mount rod bracket over each of the marked points and pencil in the screw hole positions on the brackets.
Predrill a hole at each of the screw hole points, sized for screws long enough to go about 3/4 of the the thickness of the cabinet sides. If the cabinets have sides thinner than about 3/4 of an inch, use screw anchors in each of the holes to hold the screws securely.
Fit the clothes rod into the brackets. Slide the brackets and the rod between the cabinets until the brackets are in position and all the screw holes on the brackets match up with the pilot holes in the cabinet sides. Insert a screw anchor into each hole if using them.
Secure the brackets in place with screws through the pilot holes, using a screwdriver or a screwdriver bit on your drill.