The belief in Manifest Destiny was only an incidental cause of the division of the nation. The real cause of the division was the issue of slavery and the differences it caused between North and South.
Manifest Destiny was the idea that the United States was destined by God to enlarge its territory to a significant degree. The idea did not explicitly state how much territory the US deserved to have. However, it did state that the country deserved to be larger because it was superior to other countries. It was superior in its religion because it was a Protestant country. It was superior in its politics because it was a democracy. It was superior racially to any country not made up of “Anglo-Saxons.”
This idea in itself did not really divide the nation. The vast majority of people believed in it. It did cause problems, however, when combined with the issue of slavery. The belief in Manifest Destiny caused the US to expand. For example, it brought about the war with Mexico that got the entire Southwest for the US. This caused problems because it led North and South to disagree over whether the new territory should be free or slave. This disagreement is what really helped bring about the division of the nation and the Civil War.
Thus, the idea of Manifest Destiny facilitated the division of the nation but did not cause it.
"Manifest Destiny" held that is was obvious--"manifest"--that the fledging United States should rightfully expand across the entire North American continent from the Atlantic to the Pacific and possess all the land in between (at least up to Canada). Those who believed in Manifest Destiny believed it was God's will that white Christians should control this land mass. While this fueled western expansion, the expansion itself ran into trouble over slavery: should the new territories be slave or free? Abolitionist northerners very passionately believed slavery should not be allowed to expand into the new territories and fought it as a sin. Slave states, fearing their ability to own slaves would be voted out of existence should the new territories come in as free states, fought to make them slave states. Various compromises, such as letting the people in the territories themselves decide, or bringing one slave state in for every free state, led in some cases to bloodshed and exacerbated tensions that were already running very high. In particular, the Kansas-Nebraska Act of 1854, allowing settlers the right to decide the fate of Kansas, led to violent, bloody clashes in the late 1850s as abolitionists and pro-slavery groups descended on the territory and fought for control of the state. The seeming impossibility of compromise as the country expanded west under "Manifest Destiny" was one factor that led the South to the conclusion that the only solution was to secede and form a separate country.