There were two main factors that allowed the British to feel as though they had the right to take land away from the Indians. 

The first reason had to do with what we might call ethnocentrism today.  What this means is that the British felt that they were superior to the Native Americans.  The British were, by their own lights, more civilized than the Indians.  They were Christian while the Indians were pagan.  These factors meant that the British were better than the Indians and were justified in displacing them.

The second reason had to do with ideas about the ownership of land.  The British believed in the idea of individual ownership of land.  Each piece of land belonged to an identifiable person and was generally being used in some visible way.  This was not the case in the New World.  The Indians did not practice individual ownership.  Much of their land, particularly since European diseases had devastated their populations, looked unused to English eyes.  Therefore, the British felt that the land was unoccupied and open to be claimed by anyone who could do so.