How did the people of Massachusetts react to the territory that is now Maine being split off from Massachusetts?

TL;DR: Maine's desire for self-governance arose independently of the Missouri Compromise, and was only tied to the statehood of Missouri due to the politics playing out in Congress at the time.

Maine residents had long felt distant and alienated from the government in Boston, and the farmers who settled in the interior of Maine after about 1800 were particularly inclined to such feelings. From Mass Moments:

[A]fter Shays' rebellion caused such havoc in western Massachusetts, Maine merchants worried that they would need the power of Massachusetts if they were ever faced with a similar uprising. They decided to express their grievances rather than push for statehood. But with the continued growth of the population outside established coastal towns, frustration spread to inland farmers. Unlike the coastal merchants, these folks had no common interest with the men who ruled Massachusetts and Maine. By 1800 these inland farmers were leading the drive for statehood. Their list of grievances was long, and their egalitarian, fiercely democratic politics made them eager to break with Massachusetts.

Maine held referendums on statehood in 1792, 1797, and 1807; however, in these instances, Mainers decided not to split from the rest of Massachusetts. The real breaking point was the War of 1812, which saw British forces occupy a large portion of eastern coastal Maine. From the Portland Press-Herald:

Regional solidarity collapsed in the summer and fall of 1814, as British forces surged down the coast, occupying Machias, Blue Hill, Castine and Belfast, looting Hampden and Bangor, and setting fire to a Biddeford shipyard. Residents of Wiscasset expected the village “would be laid in ashes” at any moment, while thousands of militiamen rallied to defend Portland from the expected assault.

...

With the federal government already bankrupted by the long war, Mainers and the White House alike looked to Massachusetts to take action to defend southern Maine and liberate the occupied zone. Instead, legislators in Boston chose to do nothing, while Gov. Caleb Strong carried on secret diplomacy with his British counterpart in Nova Scotia, hoping to secure assistance in the event the Bay State made good on threats to secede from the United States. When Maine’s William King met with Strong to discuss an expedition to push the British out of Maine, the plan was promptly leaked to Boston newspapers and thus to the British.

In a referendum in July 1819, the people of Maine voted overwhelmingly (over 70% in favor) to secede from Massachusetts and form a new state. A constitutional convention was held in October, and the Massachusetts state legislature passed a bill granting statehood later that year. The Massachusettsians (?) generally disapproved of this notion, but agreed to abide by the will of the people:

It was universally recognized that the decision rested entirely with the people of Maine, and there was no attempt at or suggestion of bullying them. But they were appealed to strongly to remember the glories of the State which had been won by them in common with the citizens of Massachusetts proper.

However, to be admitted to the Union, Congress and the president had to pass a bill allowing Maine's statehood as well. This is where the Missouri compromise came into play; Missouri's statehood had been hotly debated throughout 1819, with abolitionists strongly opposing admission of a new slave state. When Maine's statehood bill came before Congress in 1820, the Senate decided to link the two measures. Since the balance of power between free and slave states would be largely unaffected by one free state and one slave state joining the Union, the opposition to Missouri's statehood lessened and the bill eventually passed.