In the course of the Torah's description of Moshe's life in Shemot, it says, "It happened in those days that Moshe grew up and went out to his brothers and observed their burdens, and he saw an Egyptian man hitting a Hebrew man of his own brethren. He turned this way and that way and saw that there was no man, so he struck down the Egyptian and hid him in the sand" (ב:יא-יב). Rabbi Chaim Shmuelevitz used to say that sight is the way by which one is able to feel sympathy for others who are in pain. Just looking and turning is not merely enough. Rather, one must make a special effort to observe a situation carefully. If a person observes another's pain carefully, one will not only take notice but will also feel for that person and will have a desire to help. This is why the Sages equate one who is blind with a dead man. If one cannot see, how can one feel someone else's pain? That person is alone in the world. If one is alone in the world, then one is missing a part to one's life and is not considered fully alive.
Rashi (on פסוק יא) says that Moshe made a special effort - ויצא אל אחיו וירא בסבלותם - that his eyes should see and his heart should feel the suffering of Bnei Yisrael. Indeed, Moshe took seeing to another level. An ordinary person who sees a man suffering in a hospital bed would probably give some money to his family to help them out. However, the Midrash says that Moshe lived in a palace which was full of luxuries and could have lived a comfortable life. When Moshe found out who he really was, he left his luxurious palace every day to visit his brothers who were hard at work. He pretended that he was helping one of the Egyptian taskmasters. Inclining his shoulders, he carried the heavy burdens with Bnei Yisrael. Moshe cried for them and said, "Your tortures hurt me, too. I only wish I could die for you." This is also why during the Shacharit Amidah of Shabbat, we say, "ישמח משה במתנת חלקו," meaning that this was his happiness, that he gave his soul for Bnei Yisrael. If a community is suffering, one should not say, God forbid, "Their suffering is not my problem. I will go home, eat, drink, and make myself comfortable." Rather, one should get involved and share in the hardships of the community.
There is also another issue worthy of discussion in the pasuk. In pasuk י the Torah says ויגדל הילד - "and the boy grew up." In pasuk יא the Torah says ויגדל משה - "and Moshe grew up." The question arises, why does the Torah seem to repeat itself? Rashi says that the first time the Torah tells us Moshe grew up this meant he grew up physically. The second time was meant to inform us that Moshe rose to power in Paroh's house.
Moshe was not like most people. Most people forget their origins when they grow up. When people rise to power, they often forget their own brothers and relatives and even forget where they came from! A person in such a situation would usually ignore the suffering and hardships of his family in order to retain his own privileged position. Yet, despite the fact that Moshe grew up in the house of Paroh, Moshe did not forget his brothers. The Torah says וירא בסבלתם - "he went out to search for ways to ease their suffering."
One could ask then why did Hashem arrange that Moshe should be raised in Paroh's home? Perhaps, Hashem made it so Moshe would be raised in a palace in order for him to experience a majestic manner of behavior. He would be able to observe it closely and learn to be a leader and not a subservient personality (see אבן עזרא ב:ג). We later see that this attempt to have Moshe grow in a majestic manner did help him to develop into an active personality. Moshe killed an Egyptian in order to defend a Jew who was being beaten. He rescued the daughters of יתרו and enabled them to water their flocks. How did משה grow up to be concerned with עם ישראל despite being raised in Paroh's house? The credit goes to Moshe's mother, Yocheved. Despite the fact that she raised Moshe only for the earliest years of his life, she succeeded in filling him with love and loyalty to his people. Therefore, Moshe did not become an Egyptian prince but remained a Jew who possessed a sensitive heart and was concerned with the fate and destiny of עם ישראל.
Perhaps Moshe grew in physical strength. A small boy doesn't always understand the difference between right and wrong. Now when Moshe went out and saw the abuse that Bnei Yisrael were receiving, he took action. Previously, Moshe had gone out and felt weak, but now he had the strength to fight back. It seems frivolous to say that Moshe never saw the mistreatment Bnei Yisrael were receiving. By the time Moshe grew to a state where he could take action, he had accrued so much anger inside that he was able to kill a man.
According to some opinions, Moshe was actually 06 years old. Maybe Moshe reached the age of respect. People respect an old man greater than they respect a young man. According to the Shulchan Aruch in Yoreh De'ah, a שיבה is a man who has reached the age of 07. Perhaps Hashem gave Moshe the appearance of an old man. If Moshe appeared like he was an old man, people would give him great respect. Or the situation can be reversed. It is possible that Hashem made Moshe appear young. ויגדל משה would mean that he grew in his appearance. Hashem gave Moshe a powerful or forceful appearance.