To bench press more weight you need to develop full-body tension. Shoulder, core, and hip stability are all required to establish a stable platform from which to bench. Core stability exercises will also reduce injuries since you're less likely to get into a bad position if your core is strong. This also prevents any "power leakages" and allows for a strong, smooth press.
It goes back to the popular saying, "You can't shoot a cannonball from a canoe." If your core is weak and you don't have any shoulder or hip stability, you aren't going to be able to press big weights.
All types of plank variations and rollouts can help since they improve shoulder and lumbar stability, two crucial elements in the bench press. Adding a band around the wrists for plank exercises also "activates" the external rotators and the stabilizer muscles within the rotator cuff. Try this:
The stronger these muscles are, the more efficient your press.