Once the wrist flexes excessively, it creates a domino-effect, where the elbow and shoulder experience a greater torque. Some people, especially females, can experience pain in the wrist. However, the usual consequence is loss of adequate upper-body stability, worsening form, and therefore poor performance.
There are three reasons why people fail to stabilize their wrists. One, they just are unaware. They usually are less-experienced, and just don't know any better. This is a common scenario in novice lifters. A second reason for lack of wrist stability is weakness in the forearm muscles. Again, this is often the case for women and people who may have had wrist/hand injuries. Supplemental strengthening exercises may be neccesary. The third reason for lack of wrist stability during the bench press is lack of shoulder mobility. The wrist joint flexes more to compensate for poor mobility in the shoulder joint. While this is not as common as the first two reasons, poor shoulder mobility creates greater problems when the lifter performs an incline press and overhead press.
Learning proper set-up and cues are the keys to keeping the wrists stabilized. Start by tightly gripping the bar with your first finger and thumb. You can even use a modified hook grip where the thumb maintains pressure on top of your first finger. Wrap the remaining fingers around the bar. Once you lift-off the bar and are ready to press, keep your "knuckles to the ceiling." More specifically, point your second knuckle joint (see picture).
Keep squeezing the bar tightly to maintain stability throughout the entire set. Restabilize between repetitions if neccessary.
The video below shows all of this.