It's not so much a matter of fielding huge armies, but one of fielding those armies for prolonged periods of time at long distances from home.
The ancient Assyrians were noted for fielding large armies, and sending them hundreds of miles to undertake prolonged sieges. You don't do that without some sort of logistics and support, but we have very little information about it, aside from things like a stray mention about bringing wagonloads of extra arrows on one campaign, and claims of large numbers of carts being used to carry back loot after a few of the sieges. The fact that the carts were there at all indicates that logistics were an actual concern, but doesn't indicate whether they arrived with the army, or whether they were part of subsequent regular supply caravans.
The Greeks fielded tens of thousands of men on many occasions, but rarely more than a few days march from home, and they were often forced to return home in time for planting or harvesting.
Rome is the first case where we have any sort of detailed records about supply. Rome's protracted war and long siege against Veii, several centuries BC, required an army in the field continuously for several years (roughly 5+), which led to Rome's creation of a permanent army with regular pay and provisions. That may have been the key to many of Rome's later successes, where they lost their fair share of battles, but won most wars due to their better logistics and principle focus on strategy rather than tactics. Rome could not only put a lot of men into the field, but keep them there for as long as necessary, and could wait until the opposing army was already breaking apart due to shortages of food and/or water for the optimum moment to attack. During Gaius Julius' career, they used a combination of constructed roads, bridges, fortified supply depots at regular intervals, wagon convoys, cargo ships, trade deals with the locals, and whatever else was required to insure regular supply to their field armies.
Without that or a loose feudal system of vassals to extend your control, conquering and holding an empire more than a few days travel from your seat of government was close to impossible. You can only support so many men by requisitioning supplies from the locals as you pass, and can only carry supplies for a few days without regular resupply. With food shortages, limited water sources, and high levels of stress, disease and other problems are practically guaranteed. Only with a steady source of fresh food and water is it feasible to carry out extended campaigns with large armies.
