It was quite obvious to everybody who counted that Indians who lived in the vicinity of whites and blacks had an incredibly high infant mortality rate, etc. The only long term solution was to remove Indians to an area outside of the expanding non-Indian population. Recall, if you will, that all of this happened before the "germ theory of disease" had been developed, and more than a century prior to the development of the first antibiotics.
The initial destination was to be what is now Indiana. In fact, at that time most of what is now Indiana, and a large part of Indiana territory, consisted of swampland. It was felt at the time that most of the area was not fit for settlement in terms of ability to support modern agricultural techniques.
Then the New Madrid quake happened, Indiana pretty much drained off, and the whole territory became highly desirable for new settlement.
Of interest, long after Jackson, in the 1850s, there was a final payoff in both Chicago and in the South, where the Indian tribes moving West received money for their lands.
This was all done under treaty arrangements agreed to by the tribes.
Continuing payments have occurred as a consequence of federal "takings". For example, the BIA rented out Osage Indian land to oil companies. The Osage receive regular payments. Along the way BIA managed to lose several billion dollars of what it owed the Osage. This has happened on an even larger scale with all of the recognized tribes and Uncle Sam can be demonstrated to have absconded with tens of billions of dollars due Indians for mineral rights, etc.
There is currently a law suit in place regarding this little problem.
There was an Indian war during George Washington's Presidency which ended with the Treaty of Greenville in 1795, by which the Indians involved were forced to cede a considerable amount of territory (mostly in Ohio).
Too shamefully true.
Should be made right.