Bigger, Stronger, Lats
Not surprisingly, the one thing that very strong individuals from various sports have in common is a solid back.
The role of the lats is huge in all pulling exercises like pull-ups, rows, pulldowns, etc. That's obvious, but the lats also play a giant role in the power clean, power snatch, deadlift, bench press, and squat.
Power Clean, Power Snatch, and Deadlift
The lats keep the barbell in the proper path. Otherwise, the barbell strays from the body, giving you a disadvantageous lever to pull from and placing a greater load on the back.
To maximize performance in those lifts, you need to learn to engage the lats and keep them engaged. The sweeping deadlift drill (shown below) will help teach that, but it's not enough to just learn how to engage the lats; they need to be strong so that they'll stay active when the barbell reaches the knees. If not, the bar will shift forward and you'll miss the lift.
Here the lats stabilize the shoulder joint (if you're more stable, you're stronger and less prone to injury), give you better leverage to press by giving you something to press against (if the lats are engaged and elbows tucked in, your triceps will be in contact with your lats), and they actually help you press the bar up at the bottom of the movement.
To best engage the lats during a bench press, try to externally rotate at the shoulder joint when you set up. Or if you prefer, try to bend the bar in half. Then try to bring your shoulders down (opposite of a shrug) while lifting the chest up. Lastly, as you lower the bar to your chest, imagine rowing the bar towards you, not just resisting it on the way down.
In the squat, the lats, like the rest of the upper back, are extremely important in assuring a strong position. After all, your torso is what's holding the bar. The more solid your back is, the stronger you'll be. The barbell will feel a lot lighter and the bar path will be more stable.
To engage the lats during a squat, pinch the shoulder blades together hard and "bend the bar" while bringing your elbows to your ribs. To visualize it better, imagine doing a behind-the-neck lat pulldown with the squat bar.