Proverbs 24:16
Though the just fall seven times, they rise again, but the wicked
stumble from only one mishap.
Matthew 5:43-44
You have heard that it was said, You shall love your neighbor and hate
your enemy. But I say to you, love your enemies...
Seven is used often in the Hebrew Bible to symbolize completion. The seven days of creation is only the most obvious example. In the Book of Leviticus, the priest performs symbolic numerous acts (sprinkling, anointing) seven times. Joshua marched around Jericho seven times. Elijah sent his servant to look for rain seven times. (1 Kings 18:43) Elisha told Naaman to wash seven times in the Jordan to be cured of leprosy. (2 Kings 5) The list goes on.
In Jacob's case the number is more than merely symbolic. He bowed seven times in order to win his enemy Esau's favor and thus restore their brotherhood. Esau's heart was encrusted with layer upon layer of resentment due to Jacob's taking advantage of Esau's hunger to buy his birthright and then stealing their father's blessing, meant for Esau, in deception. One may picture each prostration as an apology and a request for forgiveness.
In this context, I also see a connection to Genesis 4:15
If anyone kills Cain, Cain shall be avenged seven times.
Jacob and Esau were in a position to restore the murder of Abel by Cain and establish a condition of unity. Abel was killed because of his elder brother's jealous resentment. This resulted after Abel's offering was accepted and Cain's was rejected. Esau inherited Cain-like resentment after Isaac blessed Jacob instead of Esau. Jacob then fled after learning that Esau intended to kill him. He now returned after 21 years, and Esau came to meet him with 400 armed men. This meant that Esau still harbored murderous intent. Jacob first sent wave upon wave of extremely valuable gifts ahead to show his sincerity. He wrestled with an angel all night and obtained its blessing. Finally, he approached Esau, bowing to the ground again and again. I picture each bow as successfully melting another layer of Esau's resentment, which was also the resentment Cain, of whom he was the representative. Whether this was literally the case or not, the bows were the final gesture on Jacob's part, representing his absolute desire to be reconciled with his brother, practicing the principle that Jesus later taught, by loving his enemy.
Secondary Questions:
Why did he bow from so far away? Because Esau had come with an army of 400 men. The text does not indicate the distance involved. It may have been only a few yards by the time all seven bows were performed. But in any case, the story shows that Esau left behind both his army and his murderous intent in order to unite with his brother.
Was Jacob advancing as he bowed? Yes. All translators indicate a progression as he bowed. Typically: "seven times, until he came near to his brother." Some make it even more clear: "seven times as he approached his brother." Jacob took the initiative, and Esau responded by running to meet him when he got close.
Conclusion: Jacob bowed seven times before Esau in order to show love for his enemy and thus resolve Esau's Cain-like historical resentment. This was the final stage of a 21-year course for Jacob, qualifying him as God's victor (Israel = Victor) and the forefather of the people of God.