Squeezing the crap out of the bar when benching works thanks to increased rotator cuff activation, better wrist positioning, and increased aggression.
Squeezing the bar sends a signal to the rotator cuff – through a process called irradiation – to pack itself. This provides more stability to the shoulder joint, which is never a bad thing when you're trying to hoist heavy loads off your chest.
Squeezing the bar can also help create a more beneficial wrist position. A neutral wrist can squeeze a bar much more effectively than a hyper-extended one.
One additional benefit of squeezing may lie simply in decreasing fear and inhibition and increasing aggression. Often the value of a cue is to shift your focus to something more important. For example, you might be nervous about trying a huge weight. Focusing on squeezing the bar shifts your attention away from that fear. It's a bit like squeezing a tennis ball if you have to go through some type of uncomfortable procedure without anesthesia.