Proving a yeast dough means allowing it to double in size. This is typically done by letting the dough rest in a warm and moist environment, which helps the yeast ferment the sugars in the dough and produce carbon dioxide gas. As the gas builds up, it causes the dough to rise and expand, resulting in a lighter, fluffier texture.
Proving the dough also allows the gluten strands in the dough to relax and stretch, which makes the dough more pliable and easier to shape. This is why recipes often call for the dough to be left to rest and rise for a certain amount of time before shaping and baking.
Eliminating air from the dough is not part of the proving process, although it can be important in certain baking techniques. Baking the dough at a low temperature is also not part of the proving process, although it may be used in some baking recipes to prevent the dough from browning too quickly. Setting the sponge is a separate step in the bread-making process that involves mixing the yeast with some of the flour and liquid to create a fermenting mixture that is then added to the rest of the dough ingredients.