How did Americans justify taking land from the Native Americans?

They needed space for "settlements".

The Native Americans didn't believe in owning land.
The European Settlers started by trading objects, in exchange for the Native Americans moving their villages.
Later, when the Native Americans were not willing to move in exchange for anything that was offered, force was used.
That was justified by labeling the Native Americans as Godless stupid heathens savages, that were not only not entitled to have the land, but to live.
Not unlike the way the modern world is treating wildlife.

I had nothing to do with it so I am clear of the charge.

They didn't, and they didn't need to. People in the colonies mostly came from Britain.. And the British were claiming land here,there and everywhere at this time. I don't think they would have needed to justify it to themselves

They justified it because most of the early settlers believed the natives to be godless savages, thus having no right to the land that they lived on and cared not for the rights and needs of the native people, beliving that there needs and wants should be met regardless of how they treated the natives.

Americans didn't do it, Europeans did. By the time we became Americans, there were hardly any left to worry about. We called it "manifest destiny" instead of rape and pillage.

They cheated the natives out of their land for they did not understand the concept of ownership.The natives had no concept of deeds and the laws of land ownership.
So the settlers offered them trinkets and other worthless things in trade of land.Later on they just killed them.So much for white civilizations that have no concept of respect for the true owners.

Racism, pure and simple.

From what I have read, arrogance.
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Since they were "heathens", they and their interests and rights were not considered to be worthy of consideration..
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Many claimed that bringing European concepts of land ownership, development, forcing them to accept Christianity etc was benefiting them. With the arrogance typical of european exploration and setlement at the time, I am sure that many believed it.

Slavery demands secure borders, lest the slaves run off to freedom. To support bondage, we committed genocide. That was the Trail of Tears.
After that the president was set. It was just what we'd always done.