The motivation for building a city was to make the builders a name (cf. Psalm 14:1). Later God would "make a name" for Abram ( Genesis 12:2-3). The object of this endeavor was to establish a center by which they might maintain their unity. [source][source][source]
"A defensive wall is the hallmark of a city (see Genesis 4:17). Cities in the ancient Near East were not designed to be lived in but were intended for religious and public purposes." [1][source]
God desired unity for humankind, but one that He created, not one founded on a social state. [1] They wanted to "empower" themselves. Both motive and object were ungodly. God had instructed man to fill the earth ( Genesis 1:28), to spread over the whole planet. [source][source][source]
The builders of the "tower" seem to have intended that it serve as a memorial or landmark, among other things. It was probably a ziggurat used for religious purposes. [source][source][source]
"Mesopotamian religion claimed that their cities were of divine parentage. A symbol of this obsession with divinity among the Mesopotamians was the ziggurat (Akk. ziqqurratu) that was erected as early as the third millennium B.C. The ziggurat was a step-ladder edifice, made up of mud bricks, whose bottom was square or rectangular. The precise meaning of the structure is unknown, though it is widely agreed that it formed a stairway between the gods and earth (cf. Genesis 28:12). At the foot of the ziggurat as well as the pinnacle was a temple area serving as a habitation for the god. Ziggurats may have been considered an earthly imitation of the heavenly residence of the gods." [3][source]