The people who came together to build the "tower of Babel" in Genesis 11:1-9 were a united group that did not want to be separated. God's negative reaction to their project implies that they were in rebellion against God. However, the text of Genesis 11 does not explicitly say what the problem was.
One explanation, based on Ancient Near Eastern background, is that this group was building a ziggurat in order to try to control and manipulate God.
Another explanation, one with a long tradition, is that this group was trying to launch an attack on heaven itself, saying like the king of Babylon in Isaiah 14:13, "I will ascend to heaven; above the stars of God; I will set my throne on high." If this was their purpose, then they were sadly mistaken. God and his divine council had to "go down" (verse 7) in order to examine their puny efforts.
A third explanation, due to the nineteenth century Jewish commentator Netziv, is laid out by Judy Klitsner in her book Subversive Sequels in the Bible. Netziv saw the group at Babel as a totalitarian state that imposed a rigid uniformity on its people. It didn't want anyone to leave, and it didn't want any new ideas to come in.
One thing supporting this reading is a parallel between Genesis 11:3-4 amd Exodus 1:10. In Genesis 11:4, those at Babel say, "Come, let us build ourselves a city....lest we be dispersed." In Exodus 1:10, Pharaoh tells his people, "Come, let us deal shrewdly with them, lest they multiply..." This "Come, let us, ....lest....." construction appears in the Bible only in these two passages.
In Exodus 1, Pharaoh enslaves the Israelites, and they make bricks and built cities, as in Genesis 11. This parallel leads to the idea that in Genesis 11, the people are being enslaved and forced to make bricks and build a city. The group at Babel is trying to make a name for itself (Gen 11:4), but it is a collective name, with no individual names mentioned. In Netziv's reading, this group is a collective like the Borg in Star Trek: The Next Generation. One midrash on Genesis 11 says that when someone carrying bricks up the tower fell to his death, people mourned for the loss of the bricks rather than the loss of a human life. At Babel it was the collective that mattered, not the individuals.
In this reading, we can see why God would put a stop to the group's efforts. God created people in his image to be able to express their individuality and make free choices. In particular, we can use our free will to seek a relationship with God. The group at Babel was cutting itself off from any relationship with God.
At Pentecost in Acts 2, an event that is kind of a reversal of Babel, the disciples of Jesus were unified, being "all together in one place" (verse 1). Their unity was not a coerced uniformity, like the one at Babel. Rather, they were united in love by the Spirit of God. That unity is expressed in a diversity of spiritual gifts, as described in 1 Corinthians 12.