The city of Rome was especially suitable for the centre of their See, since it presented them with ideas and materials and, as it were, a predetermined layout with which to work. The ideas came from the inspiration provided by reading surviving ancient literature and comparing the contgent with the remains they saw around them. And because the population of Rome was much reduced since imperial times (Rome was not to fill her walls with people until toward the end of the 19th century), there were plenty of antiquities to study, and enormous quantities of ruins which could be used as quarries for building materials, all helpfully cut into columns (usually of a well understood variety of dimensions), bases, capitals, entablatures, and the characteristic brick or tufo of the buildings covered with acres of marble veneer.
Unsurprisingly, the Papacy dressed in an imperial manner, thought of their newer empire, and acted in an imperial manner - they even had "imperium" in one of the closer antique meanings of the term, raising and even leading armies in the field. In Hobbes' superb phrase, "The Papacy was not other than the ghost of the deceased Roman Empire, sitting crowned upon the throne thereof."
Popes were celibate, and were not supposed to have children; but they were usually drawn from the aristocracy of the Peninsula (more rarely from Spain), and they had nephews who profited from their elevation to the Chair of Peter. Again, the Chair of Peter often went to members of the same family (EXAMPLES), so semi-dynasties were formed, frequently with the sophisticated building urges found elsewhere in the aristocracy. Because of the enormous costs required for building schemes, "Dip not thy finger in the mortar", was Elizabethan advice in England; but Papal families frequently had large incomes which they could direct at schemes which rebuilt Antiquity or at least imitated it, frequently with re-used antique materials as noticed above.
Luckily for our enjoyment of Rome, then, some (not all) Popes embarked on grand schemes intended to leave their mark - to stamp their fame, their gloria - on the City. (Rome is indeed The City - the one and only city, Urbs - matched as is surely decent only by Constantinople, with its Greek-derived modern name, Istanbul meaning "To The City".) There are, surely, many less productive ends to which such wealth might be put; and we should try and understand the consciousness of history and their place in it which impelled extensive building work.
Many schemes involved turning the City from a muddy, insanitary and crime-ridden mess into a paved, watered and beautified set of roads and squares. The spur for this was frequently more keenly felt in preparations for Jubilee years, when large numbers of pilgrims (and hence caqsh) might be expected. Equally, large numbers of churches were constructed, chapels endowed, and libraries bought. Water had been essential for the luxuries of ancient life; and today, thanks to the example of the antique so readily followed by the Renaissance and Baroque popes, Rome still has a profusion of fountains of drinking water (the connoisseurs claim to be able to tell from which aqueduct the different flavours of water descend).