How did Uncle Tom's Cabin caused war?

The book didn't actually cause the war; the reasons were much more complex than just the issues addressed in the book, but the book did contribute to the tensions leading up to and during the war. The book helped bring awareness of some of the worst things about slavery and helped polarize opinions on the matter. Abolitionists gained support for their cause. Slave holders felt more threatened.

Beyond slavery (which was a major issue in the war) were the issues of state's rights, import tariffs that helped northern merchants while hurting the agrarian south, and suppression of Southern influence in the legislature and in the electoral college by the greater number of representatives in congress from the North.