What is the purpose of the Tower of Babel?
Why do you think they wanted to build it?
פן נפוץ LEST WE BE SCATTERED ABROAD — That God shall not he able by bringing some plague upon us, to scatter us from here.
וסוף דעתם לבנות עיר גדולה למושבם ולבנות מגדל גבוה להיות להם לאות ולשם ולתהלה לדעת מקום העיר להולכים חוצה כרועי המקנה גם יעמד שמם אחריהם כל ימי המגדל וזהו שאמר הכתוב ונעשה לנו שם:
They wanted to build a great city to dwell in, and a very tall tower to ensure their fame and glory and to serve as a sign indicating the place of the city to those outside it, such as shepherds. The tower, as long as it stood, would also perpetuate their names after their deaths. This is the meaning of and let us make us a name.
ויאמרו הבה נבנה לנו עיר, this was at the advice of their leaders who wanted to enthrone Nimrod as king over the entire human race,
Thus we will make for ourselves a name. This “name” was the idol that was to be placed in the tower. They hoped that on account of the grandeur of the tower and the city this idol would come to be recognized universally as the supreme deity. In this way the king of the city would achieve dominion over the entire world.
So that the name of the tower and the city would go out to far places. If we go to a far away land we will know how to return here. All those who are asked about the way will know, because everyone will talk about it and it will be visible from far away.
ונעשה לנו שם, “and let us make a name for ourselves.” They referred to establishing a reputation of grandeur vis a vis any other creatures.
פן נפוץ, “lest we scatter.” They were afraid that war might break out among them, assuming there were competing nations or human beings in distant areas unknown to them. At any rate, they realised that strength lay in numbers and in visibility over vast distances, such as the top of a tower that can be seen from afar. Basically, they felt that if they were to obey G-d’s command “to fill the earth,” this would contribute to their becoming ineffective in competition with G-d.
And its top in the sky - It is obvious that you should not think that this would be one city to the whole world, rather they thought that the other nearby cities would be subjects to that city that had the tower, that they would be able to see from it to the distance, over all their settlement, so that no one would be able to separate themselves and go to another land. That's why they needed its top reaching the sky.
ונעשה לנו שם, they meant that as a result of having a permanent residence instead of being nomads, they would acquire a far greater reputation of being people to be reckoned with. If they were to travel, and when asked would refer to this city as their home town, this would impress people over the globe. Not only that, the fact that the traveler himself compared his home town with whatever other town or settlement he would encounter, would remind him of his home town comparing favourably with the place he had just visited, and he would be sure to return home. It would act as both a physical and psychological magnet to draw the traveler back home.