A strong core is the backbone of rock climbing. It not only helps to sustain you on the climb but also helps your body and muscles work together. Your core is the center of balance and your stabilizing force. A strong core keeps you anchored to the wall.
Let's get this straight - "core" does not mean abs. The core consists of all the muscles between the top of your thighs and your shoulders. These muscles are used for a variety of jobs, from breathing to helping you sit upright, to linking the muscles of the arms and legs in the "kinetic chain." This connection is what we're concerned with training. The core is critical to climbing not so much in initiating movement, but in preventing it. (from climbstrong.com)
These links below offer some helpful tips for growing a stronger core.
1. 5 Exercises for Developing a Stronger Core
2. Core Training for Climbers
3. Drills to Build a Strong Core