The short answer is: with concrete, and the cheap labor supplied by slaves and off-duty soldiers.

I talk about Roman concrete in this answer. Thanks to geological good luck (a lot of volcanic sand near Rome) and a penchant for large-scale public works, the Romans had both motive and opportunity to become pioneers in the use of concrete. This made possible not only domes and vaults of unprecedented size and durability, but also exceptionally durable load-bearing walls in buildings of all types.

The emperors (responsible for virtually every important building project in and around the city of Rome) controlled thousands of slaves, who were regularly employed in construction work. Outside the capital, much construction was carried out by Roman soldiers, who spent much of their time building/repairing roads, supervising the design of bridges and aqueducts (military engineers were the best in the business), and occasionally facilitating the building projects of local elites.

All this activity was underpinned by the emperors' massive investment in public construction. Particularly in the early Empire, when they had a special connection with the city of Rome, emperors felt obligated to advertise their generosity, and outdo their predecessors, by building amenities and monuments that would serve and glorify Rome. In the provinces, local elites, enriched by the peace and prosperity of the Pax Romana, played a similar political game by beautifying their cities with public buildings.

Once the Empire weakened and (in the West) fell, and neither emperors nor local elites had the resources to sponsor such large-scale building projects.