The numbers you might read about, like with any historical numbers, should be taken with some circumspection. But it is true that the Ottomans were able to field large armies for their time. In terms of manpower, the empire had two principal military bodies: the kapikulu or household troops and then the timariot army until the seventeenth century. The first one is pretty simple. The kapikulu consisted of the janissary regiments in the capital and those deployed to the provinces plus the six standing cavalry regiments also based out of Istanbul. The janissaries were conscripted from the devshirme initially, but by the end of the sixteenth century more and more Muslim-born men join the corps voluntarily. The janissaries had tremendous growth at the end of the sixteenth century because of the war with Persia (1578-1590) and a major war with a Habsburg led coalition (1591-1606). During this period the janissaries and household cavalry grew by about three times in number.

The main advantage they had during the fifteenth and sixteenth centuries was the structure of the timariot army. A timar was a semi-feudal land grant to an Ottoman notable, but unlike in Europe it was not hereditary. Each timariot sipahi brought himself as a mounted soldier plus a retinue to war, the size of which was calculated based on the land he had the rights to. Typically, it was these cavalry plus their retainers that made up the high numbers of Ottoman armies during their main period of conquest and expansion. By the early seventeenth century because of economic and political factors the strength of the timariot army declined. Many of these timar holders could no longer afford to outfit a war party.

Making up for the loss of timariot forces in the seventeenth century were provincial governors and their private mercenary armies. During the instability of the 1590s are early 1600s many peasants became bandits or mercenaries to make a living. By the mid-seventeenth century, these kinds of mercenary musketeers (sekbans) matched or exceeded the numbers of janissaries in Ottoman armies. I should add that research into this topic is somewhat emerging and not as well-defined as the Classical Period timariot army, so precise details are a little thin.

The overall trend for the Ottomans was a mix of centralized standing forces augmented with manpower pools they could mobilize during wartime, whether timariot or sekban. This gave them an edge over European armies of the early modern period which consisted entirely of seasonal levees or mercenaries (barring some exceptions of course). This gave the Ottomans a lot of manpower and the federated structure offset some of the costs on the state treasury.

Some additional readings:

Rhoads Murphey, Ottoman Warfare 1500-1700

Baki Teczan, The Second Ottoman Empire: Political and Social Transformation in the Early Modern World