We can't allow the admission of another slave state. Adding a slave state will spread slavery, but adding a free state won't limit it.
Northern Free States Beliefs about Congressional Power
The laws of congress are final, and are above the jurisdiction of the state governments.
Northern Free States Fears about the Compromise
This compromise will spread slavery further west.
Southern Slave States Why did they support it
Many northern abolitionists wanted to stop the spread of slavery altogether, end of the Missouri Compromise would not only admit a new slave state, but continue to allow further admission a slave states in the southern half of the United States.
This is one step closer to getting states like Texas.
Southern Slave States Beliefs about Congressional Power
Northerners believed that the federal government should be allowed to regulate slavery, since it is the highest level of government in the United States.
The States are sovereign, not Congress, for they joined the union by choice, and this puts state laws above those passed by Congress.
Southern Slave States Fears About the Compromise
The abolitionists of the northern states dislike the idea of a compromise that would allow slavery to, not only continue to exist, but continue to spread west as well.
Now we'll be unable to spread slavery further north, and there's only so much more to the west.
The South supported the compromise because it would allow them to admit Missouri as a slave state, and thus spread slavery further west.
The South believed that America wasn't as much of the nation as it was a union of independent States, and that this union was simply an agreement between the states, like a treaty. Thus the south believed that states had rights such as nullification and secession.
The South could no longer spread slavery North of the 36 30 line of latitude, and had limited amounts of western land below it.