When the Mexican-American War ended, the United States received the Mexican Cession which included all of California, Nevada and Utah, and parts of Arizona, New Mexico, Colorado, and Wyoming. With this huge land acquisition, the issue of slavery in the new territories was raised. This led directly to the Compromise of 1850. This compromise deepened the division between the North and the South. First, California would be admitted to the Union as a free state, which upset the South. Two new territories would be established, Utah and New Mexico. These territories would be open to slavery through popular sovereignty, that is, the people of the territories would decide the issue of slavery. This upset the people in the North because so much territory could now have slavery. The slave trade was banned in the District of Columbia which upset the South because they saw this as a first step towards abolishing slavery. There would be a new, strict, fugitive slave law which upset the North because now by law they had to assist in returning slaves to their owners. These tensions created by the Compromise of 1850, which was passed because of the results of the Mexican-American War, helped create the conditions for the Civil War.
Before the war with Mexico, the North and the South had the issue of slavery more or less under control. The Missouri Compromise had settled which areas would have slavery and which would not. Of course, there was still tension between the two sections over slavery, but they were not at an extremely high point. The war with Mexico changed that, making the tensions much more intense.
The reason for this is that the war with Mexico led to the taking of huge amounts of land. The US had to decide what the status of slavery in these new territories would be. This meant that the whole issue had to be debated which, of course, led to tensions as the North and South each tried to get their way.
Thus, by forcing Congress to decide on the legality of slavery in huge new areas of land, the Mexican-American War intensified sectional conflict.