According to Megillas Sefer, Haman was saying to Esther that if she forgives him, Achashverosh will, too.
The Vilna Gaon writes that Haman tried to tell Esther that he didn’t know that the Jews were her people.
The Meshech Chochmo writes that Haman realized that the invitations came from Esther, so she is the one with the most power.
According to the Ben Ish Chai, Haman tried to convince Esther that he put her in this position of power by getting rid of Vashti, so she owed him a favor. The Malbim posits that perhaps Haman would not have approached her under any normal conditions, but she was the only one left, so he tried his alternative (Plan B) excuses on her. Seeing that she is a woman, and particularly a Jewish woman, he was hoping she would show Haman mercy.
The Sfas Emes writes that Esther fought the urge to be merciful, unlike Shaul with Agag. She accomplished this by having been exposed to Haman. In this way, she emotionally hated what she was intellectually commanded to hate. Similarly, first the verse (Devarim 25:17) commands us to remember Amalek, and only then (Devarim 25:19) to destroy it. First, one is required to have the emotion, and then to perform the act.
The Sfas Emes points out that, on a spiritual level, this act of Esther’s was a tikkun (“repair”) for Shaul’s error of allowing Agag to live. The Zer Zahav adds that Haman’s begging was a great test for Esther’s sense of improperly placed mercy. After all, the Rambam (Mishneh Torah, Hilchos Teshuva 2:1) writes that the ideal way to demonstrate complete repentance is to be faced with the same challenge, and to nevertheless overcome it, and this was almost a direct parallel to the story of Shaul and Agag.