I'm going to have to disagree with FuzzyChef's answer a bit. I don't understand the statement that FuzzyChef's wife's recipe matches The Spruce Eats (separating egg recipe) proportionally, but Pepin's supposedly uses more eggs. Yet The Spruce Eats recipe uses 2 eggs with 1/2 cup milk and 1 1/2 tablespoons each of butter and flour. Pepin uses 5 extra large eggs (or about 6 large eggs) with 2 cups of milk and and 6 tablespoons each of butter and flour. That is, Pepin's recipe is scaled up 4 times in most ingredients from The Spruce Eats, but only about 3 times in eggs. Pepin is thus proportionally using less eggs than The Spruce Eats recipe.
And while I agree that Pepin does beat the eggs in a manner that incorporates some air, that's clearly not what's causing the batter to rise, as he then follows up by noting that one can prepare this souffle batter hours or even days ahead. Most of the small amount of air incorporated by beating the whole eggs will almost immediately dissipate and will mostly be gone within minutes, let alone hours or days. Whole eggs, unlike egg whites, do not hold their foaminess for long. The beating is helpful, but clearly not the essential process to the rise.
Instead, the science behind this is quite simple and truly basic: the souffle rises because of steam.
The reason this technique works is the same reason that popovers and Yorkshire pudding rise (to a huge extent, often much more than classic French souffles), with no egg separation and no fancy preparation. But the bubbles created by the steam in popovers and Yorkshire pudding are generally large, creating a more "rustic" texture, rather than the refined souffle. Pepin's dish has a less extreme rise and more consistent texture because he puts a large quantity in a wide thicker pan, rather than, say, popovers, which are placed in relatively small narrow pans that heat the batter very quickly (and thus produce incredible oven spring from the steam).
Once the steam begins to rise, the eggs begin to coagulate and trap it, leading to the light and airy texture. The flour also helps provide more structure to hold up the finished product. Essentially, it's the same steam that makes light and "fluffy" scrambled eggs, just increased by more moisture and trapped with the help of flour to provide additional structure. Popovers and Yorkshire pudding proportionally contain a lot more flour, which allows them to rise much higher and have a more "bread-like" texture, rather than the smooth texture of a souffle. But they're both using the power of steam primarily.
The classic French souffle technique with egg separation isn't necessary to create a rise -- steam can do that all by itself. (Harold McGee in On Food and Cooking notes that the air bubbles from the egg-white foam only contribute about 25% of the air to a souffle: about 75% of the air comes from the steam produced during baking.) The French souffle with the egg white beating, etc. is about creating a very consistent fine-grained interior texture (with small bubbles), all produced by a relatively slow and consistent rise in the oven. In Pepin's case (and in most standard souffles), cooking the flour and adding cheese help to thicken the batter, so less flour is needed to start trapping air compared to things like popovers. And Pepin's reduced egg quantity also contributes to a slightly thicker batter that won't rise as high, but is also less fragile and less likely to fall. The longer bake time in a wider mass contributes to a more slow, even rise, where the top layers gradually coagulate and help trap air in the interior.