Microbial (also called “biological”) leavening is the grandaddy of all leavening techniques. It dates back to when the first human mistakenly left his bowl of gruel sitting out while he rushed off to join the village mastodon hunt. The result was what we now call a “starter”. And while the biology of microbial leavening can be a bit complex, mechanics are pretty simple: yeasts, as a by-product of their consumption of sugar, create bubbles of carbon dioxide in a dough. Those bubbles fill with steam in the oven, expanding the loaf and lightening its texture. Elementary, my dear Watson.
For more much more on microbial leavening have a look at the yeast section under Baking Ingredients > Leavening Agents, and especially the entries on starters. (Note: much more information that these links can be found using the left-side menus).