The Puritans believed they were part of God's elect, those lucky few predestined by the Almighty to go to heaven. This gave them a sense of moral superiority over those they regarded as predestined for hell. Such hell-bound sinners included Native Americans, whom the Puritans damned as pagans.

Simply put, the Puritans believed they had God on their side in establishing themselves on American soil and that this therefore entitled them to treat the Native population however they pleased. As far as the Puritans were concerned, America had been given to them by God for the purpose of establishing a godly kingdom on earth; as the Native Americans weren't considered godly, there was no place for them in this shining city upon a hill. The Puritans, then, had no compunction with taking land that didn't really belong to them. In stealing what belonged to the Native Americans, they told themselves they were doing the Lord's work.

And God blessed them, and God said unto them, Be fruitful and
multiply and replenish the earth, and subdue it: and have dominion over the fish of the sea,
and over the fowl of the air, and over every living thing that moveth upon the earth.  --Genesis I:28. 

The Puritans believed that God blessed them with the lands of the New World. Their main justification for taking Indian land was that the Native American populations were not using the land effectively, so it was their divine right to take the lands that belonged to the Native Americans.  Since the Native Americans did not understand land ownership in the same way that the English did, it was not as difficult to take it from them.  The Indians did not build cities or towns and the land looked to be undeveloped in the eyes of the Puritans.  The colonists believed that the land was being under-utilized and that they were actually providing the Indians with a service by civilizing their culture.  In quoting scriptures from the Bible, like the one referenced above, the Puritans felt a sense of moral calm in taking the land from the Native Americans.