First let's consider what the old revolutionary ideology towards the native people's. Revolutionary belief envisioned the native people's gradually becoming "civilized' and becoming farmers. This policy would have resulted in the loss of the vast majority of native land and the complete destruction of native culture. Jackson actually coached his justification for Indian Removal partially on these grounds, pointing out that the native people's of the North East had been assimilated into white society. Removal to the west would thus protect native civilizations in a sadly ironic twist of fate maybe he was even partially correct. Prior to Jackson administrations had already been encouraging native people's to immigrate west, particularly during Calhoun's tenure as secretary of state. These movements were voluntary however, and efforts to assimilate the native people's continue. White rule in the early Republic was far from benevolent however, and in my opinion Indian policy from the Early Republic to Jackson has far less dramatic changes than some Jackson opponents understand. The Early Republic was full of wars that stripped the native people's of huge swathes of territory and forcibly removed them from their lands, such as the treaty fort Jackson. But on to why Indian removal became such as issue.

Andrew Jackson's basis for Indian Removal is complicated to say the least. To start off he was very angry and distrustful of the Federal government after Washington's Indian policy. Washington's policy had the vast majority of the Federal army deployed against the natives of the Northwest, by contrast troops deployed to the Southwest were far smaller. Jackson had also participated in a large number of wars in which foreign nations had utilized the native peoples of the United States to wreck havoc: The British had supported numerous Indian nations during both the revolution and the war of 1812, Spain had supported the Creek Confederacy and later supported the Seminoles from Florida. Removing the tribes to the west would thus serve as a safety precaution against future conflict. Jackson's administration's type priority was also dealing with the nullification crisis as the number one priority and every other issue was of secondary importance. Endorsing state Indian removal, along with the Maysville road veto, helped ensure that other state's rights advocates wouldn't side with South Carolina.

Georgia was the most important state when it came to Indian removal and had a very unique past in the dealings with the Federal government. It would take a lengthy post to go into the full details but the Yazoo Land Fraud was basically a scandal in which the Federalist controlled state government and sold a lot of land to speculators at an extremely low price. When the Republicans came to power in Georgia they repealed the sale, and angry speculators took the issue to the Federal government which eventually ruled in favor of the speculators ( very TLDR version). The scandal along with the before mentioned neglect of the Southwestern frontier by Washington helped inspire a healthy amount of distrust of the Federal government by Georgia. On the national level the last two years of Madison's administration, Monroe and Adams had lead to a dramatic strengthening of the Federal government the establishment of a powerful navy, BOTUS II, the first protective tariff, pensions to veterans, debates on the exclusion of slavery from the west all contributed to a rising fear and eventual backlash from state's rights advocates.

A few other notes, the most important being that a regular theme throughout American history is the drive west by the people, the government's lack of strength to stop them, and eventual conflict between colonists and native people's. Whether under British rule or American rule, the central government lacked the forces to stop the growth into the west and it was a frequent boiling point between the people of America and whichever central government was in power, be it during Bacon's rebellion or 1820's Indian removal. For instance Washington only assigned around 100 regulars to keep the peace in the SW between the creek and the westerners, a woefully inadequate force. In no time Americans had violated the terms of the treaty and the Creek went back to being supported by the Spanish, aggravating relationships in the South West. In regards to the immediate concerns of the 1820's and 1830's, Georgians wanted more land for cotton ( as well as some gold that had been discovered) and the Cherokee had resolved not to sell anymore land.

Edit: I realize I discuss many different events and people in this post, often in very short version. Feel free to ask for clarification regarding any points brought up that are not understandable