There were two main reasons that Assyrians engaged in this practice. Note also that the Assyrians were not the only civilization that did this, they just did it more barbarically than most.
1) People-Land Connection: In the Ancient Period, people felt a strong connection to land upon which they and their ancestors had lived. It was from living on that land that they often determined their identity. This contrasts with the modern nationalist or patriotic view that identity is determined by culture, history, ethnicity, and beliefs, but not land. (I.e. in the Ancient Period you were Aramean because you lived in Aram; if you no longer live in Aram, you are just a foreigner.) Therefore, relocating people to lands in which they had not previously lived was an effective way to sever cultural histories of conquered nations.
2) Religious or Tribal Revolt: In addition to moving the people of one area to another, they mixed different hostile groups together and separated parts of the same original group. This made the residents of the new cities wary of one another as their prior tribal feuds would cloud their current judgment. On account of this, the Assyrians ran less of a risk that the subjugated people would rebel in any given area.
Usually as a punishment for rebellion. Once removed, their lands would be given to people less likely to rebel.
And of course the people who were moved elsewhere no longer had ties to the lands or kingdoms that caused them to rebel in the first place.
The more drastic alternative was to kill all males and enslave the women and children, so it could be worse for the deportees.