It's necessary to maximise the weight lifted. Some force is transmitted through the trunk as opposed to through the spine and back muscles. The organs are broadly speaking incompressible, but the lungs are not.
Squeezing the abs pushes the organs up and holds them in place, compressing the air in the lungs, creating a more rigid trunk. This stops lifters from having their chest drop down, which will cause a dropped lift or even an injury.
Of course, if you squeeze the gut, the stomach can decide it's time to vomit if you have a lot of food in it. So ... don't do that.
Similarly if you're having lower intestinal disquiet.
Keeping a strong core (which, at first, requires a mental focus on tightening your abs at certain points before it's built into muscle memory) is necessary for good form with a variety of the olympic lifts.
Having stronger abs helps with a couple workouts, especially those related to balance. Easiest to notice if you're not accustomed to pushups. Blast out a set until you're exhausted, then feel your abs burn for the next few days. The stronger your core is, the heavier you can lift since you're no longer fighting your muscles to keep yourself in balance.
During training, the secret to performance is sugar. Dextrose, to be exact. The CNS can run on ketones, but it works best with ready blood sugar.