Using a foam roller can teach good lifting mechanics on a variety of movements, particularly chest presses. The foam roller allows the scapula to move freely without being encumbered or fixed to the bench. This optimizes natural scapulohumeral rhythm and glenohumeral joint mechanics similar to how a push-up or landmine press allows optimal scapular movement.
It forces you to create heightened spinal rigidity and natural curvature because anything other than proper posture will literally feel miserable on the back. However, with proper positioning, it actually feels quite therapeutic on the spine.
The foam roller has a tendency to roll and move unless you remain tight and lock in the core. This creates rotational forces that you must resist to keep from falling off the roller. Any wiggling, cheating, asymmetrical pressing, or shifting will cause you to lose balance.
It requires you to aggressively activate the feet and ankles to help grip onto the floor. In fact, screwing your feet into the floor is almost a prerequisite when doing this exercise because anything less will result in loss of balance and instability.
Lack of good alignment and activation of the foot and ankle complex during chest presses results in decreased neural drive up the kinetic chain, including reduced signaling to the upper-body pressing muscles. But activating the feet and ankles increases neural drive, full body tension, and motor control via irradiation and concurrent activation potentiation.
In other words, you'll increase force production and your 1RM. If you have trouble driving with your hips and legs during the bench press, more than likely it's related to poor foot and ankle activation. The foam roller press helps to resolve this subtle yet significant recruitment issue.