By squeezing at the peak of s lift you're essentially increasing the time under tension by making the rep last a little longer. You're slso achieving a maximal contraction for ghe muscle; if you're curling, for example, 45lbs dumbells, you're not making the muscles contract their hardest because you can curl the weight more than once. By squeezing you're forcing a maximal contraction regardless of the weight used. This is obviously better than using a weight you can only do one rep with.

As for getting a pump; it pumps more blood into the muscle, stretching the fascia and providing loads of good, muscle building nutrients. This aids building muscle. Hence why bodybuilders are bigger than powerlifters.

Basically the squeeze at the peak and the pump are methods for inducing hyper trophy; one by increasing the muscles' perceived workload the other by pumping in more nutrients. The benefits are bigger, stronger muscles.