“She called his name Moses, as she said, ‘For I drew him from the water'” (Ex. 2:10).
It is interesting that the name by which Moses is know is the name given to him by the Egyptian princess who saved his life, rather than the one given to him by his biological parents. One of the ideas given is that his whole life, Moses was aware that his survival was really a miracle and he felt an obligation to “pay it forward” and to look after others just as the princess looked after him.
That’s why Moses is incapable of seeing injustice around him without attempting to put a stop to it. His first encounter is with a Jew being beaten by an Egyptian; his second is with two Jews fighting against each other; and finally, with non-Jewish girls being taken advantage of by other non-Jews. In all three cases, Moses stepped in to intervene on behalf of a stranger and pay forward the kindness that was done for him as an infant and repeatedly ingrained in him every time he heard his name.
The confidence that he gained being brought up in the palace, in conjunction with the sensitivity he possessed to maintain the knowledge that his life was a gift, enabled him to become the leader of the Jewish nation.