R’ Avigdor Miller points out that fasting for three days is difficult, and accomplished an unprecedented amount of teshuva.
The Talmud (Yevamos 121b) uses this verse to inform us that it is difficult, although not miraculous to be without food for that long.
The Midrash (Esther Rabbah 8:7) writes that these three days corresponded with the 13th, 14th, and 15th of Nisan, which included the first day of Pesach. When questioned regarding why Pesach should be foregone, Esther pointed out that there would be no Pesach if the Jews were wiped out.
The M’nos HaLevi quotes from the Yalkut Shimoni that these three days were the 14th, 15th, and 16th of Nisan. The Ohel Moshe points out that the main difference is whether or not the Jews of Persia had the second Seder.
The Maylitz Yosher writes that the Jews were expected to fast on Pesach in order to shock them into realizing the seriousness of their predicament.
The M’nos HaLevi writes that the three days correspond to three sins regarding which Esther expects to be guilty: eat non-kosher food, submit herself to Achashverosh, and partial complicity in the death of Hasach.
Rabbeinu Bachya writes that H-Shem only challenges tzaddikim for three days. For example, when Avraham went to potentially sacrifice his son, he found Mount Moriah in three days (Bireishis 22:4). Also, when the brothers were taken by Yosef, they were imprisoned for three days (Ibid. 42:18). Furthermore, Yonah remained inside the big fish that swallowed him for three days (Yonah 2:1). R’ Dovid Feinstein writes that the three sections of the Written Law (Torah, Nevi’im, and Kesuvim) were given to three groups of Jews (Kohanim, Levi’im, and Yisroelim) for which they needed to prepare for three days (Shemos 19:11).
The Ben Ish Chai writes that the Torah affects us on three different levels: thought, speech, and action. Therefore, Esther was telling Mordechai that the Jews need to prepare these three days to perform honest repentance through thought, speech, and action.
The Ginzei HaMelech quotes the Vilna Gaon (on Bireishis 27:13) that when Rivka told the nervous Yaakov to place the blame of his upcoming deception “eilai” (“on me”), this word can be an acronym for Eisav, Lavan, and Yosef. Those may be the greatest of Yaakov’s tests in life, that came along with the blessing he gets from his father.
Also, the Ginzei HaMelech points out that these are three different types of people: Eisav represents a glutton; Lavan represents idolatry, and Yosef represents the challenge of intermarriage. These same three issues are the ones for which Jewish existence was threatened in the Purim story. Pri Tzedek quotes from the Zohar on Chukas that the three patriarchs, Avraham, Yitzchak, and Yaakov, represent three characteristics: kindness, awe, and truth. These are the polar opposites of the three characteristics which, according to the Mishnah (Avos 4:21), destroy one’s life: jealousy, lust, and honor. During these three days, then, Esther wanted the Jews to perfect themselves in these three areas.
The Ben Ish Chai points out that three days is 72 hours, and this is the gematria of chesed, (“kindness”) (8+60+4=72). Therefore, the Jews were supposed to spend these days evoking H-Shem’s Kindness.
R’ Avraham Sutton points out that 72 is also the gematria of H-Shem’s four-letter Name when each letter is spelled out with all the yuds included ([10+6+4]+[5+10]+[6+10+6]+[5+10]=72).