Why did the north and south struggle to control the territories before the American Civil War?

As of the Compromise of 1850, there were 16 free states and 15 slave states, after California was admitted as a free state. The number of slave and free states had been running "neck and neck" ever since the Missouri Compromise (of 1820), which led to 12 free and 12 slave states, and even before.

One problem (for the South) was that the Missouri Compromise limited slave states to those territories south of 36 degrees, 30 minutes (the southern border of Missouri), even though Virginia, Kentucky, and Missouri itself were all north of this line. Kansas is on the same latitude as Missouri, and in a scramble for numerical equality, every state counted. The southerners were hoping that by "rolling back" the Missouri Compromise, a whole tier of states north of 36 30 (and west of Kansas) could become slave states, including Colorado, Utah, Nevada, and perhaps even California would "reconsider." This would be in addition to modern Oklahoma, New Mexico, and Arizona, which are south of that line.

The South's deepest fear, from the early drafts of the Constitution, was that a strong central government would outlaw slavery in their states. The entire reason the USA was given a Senate where each state holds equal power was to prevent this. The fear was that if free states ever got a majority in the Senate (to go along with their presumed popular majority) they could simply outlaw slavery, or pass a constitutional amendment to do so.

Fast forward to when the country started adding states. The only way for slave states to maintain the Senate balance they felt their way of life depended on was to add slave states every time a free state was added. So any effort to add new states without adding new slave states wasn't just a theoretical political issue; in the South it was viewed as an existential threat.

This is also why when a new party (Republican) arose in the North that had a platform of "no new slave states", people in the South heard, "we are outlawing your way of life", and acted accordingly. When it won the presidency without a single southern vote, that became, "we are outlawing your way of life, and you can't stop us".

The actual trigger point wasn't a Senate Majority (which was given up in 1850), but rather a more general sense of increasing "containment" by the South. Early in the life of the country, the slave/free balance remained even because there was land suitable for both to expand.

1848 and the War with Mexico added some land, but this land had no slaves and there was a contingent in Congress that didn't want to extend Slavery - the Free Soilers. The Compromise of 1850 kept them from extending it to California, and the rest of the land out west seemed unsuitable for large scale slave importing, it being desert and waste for the most part.

Enter Kansas. By the Missouri Compromise of 1820, this was to be free soil, but the slave holding areas of Missouri were in the North, just adjacent to Kansas and didn't want free ground so close. These fellows felt surrounded, and the South tricked Douglas into allowing slavery in as "Popular Sovereignty". It just got worse when the real free soil settlers soon outnumbered the slave settlers and the Missourians who rode in to vote about 10-1, despite rigged elections and even a mini Civil War. Kansas was going to end up a free state regardless.

This political disaster elevated Free Soilers (and others) into a new major party, the Republicans, dedicated to locking slavery out of any new territory. The strength of this party in the North and the fact that many Northern Democratic voters weren't interested in new slave states either meant a compromise that might result in a new slave state was out of the question going forward.

In 1857, one last stab at 'escape' was made with the Dred Scott decision by the Supreme Court, which purported that Slavery could not be restricted anywhere, in the Territories nor in Free States. This absurd conclusion, which in essence tried to make the Republican Party unconstitutional and make all states slave states, assured the rupture of the Democratic Party and the win by Republicans in 1860.

At that point, those fighting containment could only do it outside the US. The radicals managed to pull some states out, and then induce a war that would get nearly all the slave states to follow. They were terrified that unless it came to arms, a compromise like those in 1832 or 1850 would follow.

It was about preventing "free" states from gaining a majority in the Senate.

When the United States was formed, slave states outnumbered free states 8 to 5, giving them a majority in both the Senate and the House of Representatives. Over the course of the next twenty or so years, states were admitted or outlawed slavery such that an even balance (9 slave states, 9 free states) existed in the Senate, while the population shifted to give free states a majority in the House.

At this point, admitting a free state without admitting a slave state to balance it would give free states a majority in the Senate, leading to the eventual abolition of slavery (or, and this is what all the compromises were intended to prevent, civil war). Since seats in the House are determined by population, it was impossible to balance the House (new states tended to be low-population, having only one Representative).

Instead, between 1816 and 1850, states were admitted in slave/free pairs to keep the balance in the Senate: Mississippi/Indiana, Alabama/Illinois, Missouri/Maine (the Missouri Compromise), Arkansas/Michigan, Florida/Iowa, Texas/Wisconsin.

In 1849, California passed a constitution that outlawed slavery and applied for admission to the United States. There was no possibility of adding a balancing slave state (the territories of the Mexican Cession were judged unsuitable for slavery, and adding a slave state north of the Missouri Compromise line would have been politically unacceptable). Instead, the Compromise of 1850 was passed, which admitted California as a free state, granted some concessions to the slave states (in particular, a strengthened Fugitive Slave Act), and staved off civil war for a decade (in 1850, the flashpoint would have been Texas rather than South Carolina).