A parable from Rabbi Israel Baal Shem Tov:
A King had an only son, the apple of his eye. The King wanted his son to master different fields of knowledge and to experience various cultures, so he sent him to a far-off country, supplied with a generous quantity of silver and gold. Far away from home, the son squandered all the money until he was left completely destitute. In his distress he resolved to return to his father's house and after much difficulty, he managed to arrive at the gate of the courtyard to his father's palace.
In the passage of time, he had actually forgotten the language of his native country, and he was unable to identify himself to the guards. In utter despair he began to cry out in a loud voice, and the King, who recognized the voice of his son, went out to him and brought him into the house, kissing him and hugging him.
The meaning of the parable: The King is G‑d. The prince is the Jewish people, who are called "Children of G‑d" (Deuteronomy 14:1). The King sends a soul down to this world in order to fulfill the Torah and mitzvot. However, the soul becomes very distant and forgets everything to which it was accustomed to above, and in the long exile it forgets even its own "language." So it utters a simple cry to its Father in Heaven. This is the blowing of the shofar, a cry from deep within, expressing regret for the past and determination for the future. This cry elicits G‑d’s mercies, and He demonstrates His abiding affection for His child and forgives him.
A parable from Rabbi Levi Yitzchak of Berditchev:
A king was once traveling in the forest and lost his way, until he met a man who recognized that he was the king and escorted his master out of the forest and back to his palace. The king later rewarded him with many presents, and elevated him to a powerful minister's post.
After a while, however, the man committed an act which was considered rebellious against the king, and he was sentenced to death. Before he was taken out to be executed, the king granted him one last request.
The man said: “I request to wear the clothes I wore when I escorted His Majesty when he was lost in the forest, and that His Majesty should also wear the clothes he wore then.”
The king complied, and when they were both dressed in the garments they wore at the time of their meeting, he said, “By your life, you have saved yourself,” and called off the execution.
The meaning of the parable is that when G‑d gave the Torah to Israel, he offered it first to all the nations of the world. They all refused, except the people of Israel, who willingly accepted the yoke of Heaven and fulfilled the commandments of the Creator.
But now we have transgressed and rebelled, like the man in the parable, and with the arrival of the Day of Judgment we are fearful indeed. So we blow the shofar to recall the shofar blowing that accompanied our original acceptance of the Torah and coronation of G‑d. This merit stands by us, and G‑d forgives us all our sins and inscribes us immediately for a year of goodness and life.