There are two main reasons why the Kansas-Nebraska Act can be seen in this way.
First, it did away with the Missouri Compromise. While the South had generally been somewhat unhappy with the Missouri Compromise, it had kept the issue of slavery in the territories off the front burner. It had allowed the North and the South to get along. Now, Douglas came along and essentially threw the compromise out the window, reopening the issue of slavery in the territories and greatly antagonizing the North.
Second, it brought about the violence of “Bleeding Kansas.” Once the Kansas-Nebraska Act established popular sovereignty for Kansas, violence erupted. People from both the North and South started moving into Kansas to try to ensure that their side would have more votes when it came time for the territory to vote on slavery. Each side committed atrocities against the other to try to drive the other side out of the territory. The violence inflamed opinion in both the North and the South.
For these two reasons, some might say that the Kansas-Nebraska Act was the single most important step towards the Civil War.