Moshe wanted Aharon to come with him to Pharaoh as not to hurt Aharon’s feelings.
I will bring you into the land which I swore to give to Avraham, Isaac, and Jacob, and I will give it to you for a possession, I ided.” But when Moshe told this to the Israelites, they would not listen to Moshe, their spirits crushed by cruel bondage. But Moshe appealed to Hashem saying, “The Israelites would not listen to me; how then should Pharaoh heed me, me—who gets tongue-tied!” So Hashem spoke to both Moshe and Aharon in regard to the Israelites and Pharaoh king of Egypt, instructing them to deliver the Israelites from the land of Egypt. (6:8-13)
It is interesting to note the dialogue between Moshe and Hashem. It began in Parshas Shemos:
And Hashem said to him, “Who gives humans speech? Who makes them dumb or deaf, seeing or blind? Is it not I, Now go, and I will be with you as you speak and will instruct you what to say.” But he said, “Please, O my lord, make someone else Your agent.” You shall speak to him and put the words in his mouth—I will be with you and with him as you speak, and tell both of you what to do.
To understand the behavior of Moshe towards Hashem, we can say that Moshe fully understood the intention of Hashem was to send both Moshe and Aharon to speak before Pharaoh. If Moshe felt that Hashem only wanted him to speak, he would have accepted the role gladly. Moshe didn’t mind that he was not a gifted speaker. Moshe was in Mideon by his father-in-law Yisro until he was 80 years old. He comingled with people and his speech didn’t bother him. Moshe’s intent was for the honor of his older brother Aharon. Of course if Moshe would have gone alone to Pharaoh and delivered the message of Hashem, Pharaoh would have understood him. Hashem puts words in the mouth of a person, and Hashem has the means of having people understand His message clearly. If Moshe would have done so, he would have become so elevated, but his brother Aharon might have felt hurt. Moshe was willing to sacrifice his elevation in order to save face for Aharon.
Hashem understood the intention of Moshe, and kept on pushing Moshe to accept the role himself. But Moshe appealed to Hashem saying, “The Israelites would not listen to me; how then should Pharaoh heed me, me—who gets tongue-tied!” Hashem conceeded to Moshe, and the Torah states: So Hashem spoke to both Moshe and Aharon in regard to the Israelites and Pharaoh king of Egypt, instructing them to deliver the Israelites from the land of Egypt. Both will go and speak to Pharaoh.
Suddenly, the dialogue is interrupted with a genealogy of the tribes of Reuven, Shimon and Levi. This genealogy includes the life-span of Levi, his son Kehos and his grandson Amram; and the family of Amram and Yocheved, including Aaron and Moshe, ending with Aaron’s grandson Pinchas (Shemos 6:14-25). Then, before resuming the narrative, we are reminded:
These are Aharon and Moshe to whom Hashem had said: “Bring out the Children of Israel from the land of Egypt according to their hosts.” These are they who spoke to Pharaoh, king of Egypt, to bring out the Children of Israel from Egypt, these are passages where Scripture mentions Aaron before Moshe, as in this Posuk, and other verses where it mentions Moshe before Aaron; this is done in order to indicate that they were equal in all respects. This shows that Hashem acknowledged the greatness of Moshe wanting to have Aharon as his partner in redeeming Klal Yisrael, since they were equal.
The intention of Hashem was to send Moshe and Aharon to Pharaoh, since they were equal in level. Yet Hashem wanted to give Moshe the opportunity to rise to a unique level. However, Moshe was the most humble person in the world as Hashem attested: "Now Moshe was a very humble man, more so than any other man on earth." He put aside his own gains in order not to have his brother Aharon feel any slight inner pain. When Hashem saw that Moshe meant it fully hearted, He agreed to send both Moshe and Aharon. But, Moshe didn’t lose out at the end, and he rose to great levels not reached by anyone else on earth. We see this clearly Bamidbar 12:7: "Not so with My servant Moshe; he is trusted throughout My household. With him I speak mouth to mouth, plainly and not in riddles, and he beholds the likeness of the LORD. How then did you not shrink from speaking against My servant Moshe!” Moshe was able to speak to Hashem, because he put aside his own honor, and Hashem admired that. Therefore, Moshe achieved his unique level which was originally destined to him by Hashem.
(Yehuda Z. Klitnick).