The M’nos HaLevi writes that the reason Esther wanted the Jews to fast for her is, as it says in the Talmud (Baba Kama 92a), when we pray for someone else to have rachamim (“pity”), we earn rachamim, too.
R’ Dovid Chadida points out that Esther is really the only person in danger now. The Jews, after all, are only threatened in a year from this point. To answer why Esther wants the people to fast specifically for her, he quotes the Rambam’s commentary on the Mishna (Taanis 1:5) that a city can only establish a fast day for itself for one day. This only applies when it is a city in danger. However, when we pray for one person, the fast can last for multiple days.
The Me’am Loez brings another Mishnah (Taanis 2:1) that fasting alone is meaningless; the point is to pray, cry ,and perform teshuva (“repentance”).
Esther’s emphasis on fasting is to help the Jews realize that attending Achashverosh’s feast was the act for which they were to repent. Assuming she did not attend the feast, then, Ginzei HaMelech asks why Esther has to fast. He suggests the Jews went to the party because, on some level, they had a lack of faith in their relationship with H-Shem, thinking they were unworthy of His love and concern. According to the Ginzei HaMelech, Esther has been displaying a similar lack of faith in herself.
Incidentally, the Ben Ish Chai writes that she asked the Jews to fast eilai, for her, which can be spelled ayin (seventy) lee (“for me”), meaning that she, herself would only fast for seventy hours, which will be explained later, G-d Willing.
In his commentary on this verse, the Vilna Gaon writes that the expressions used in this verse refer to the four components of man – the nefesh (animus, ability to move), the ruach (life force), the neshama (spiritual soul), and the guf (physical body). He writes that the nefesh and guf are really one, as according to the Zohar which writes that “the nefesh is a partner to the guf.” Haman wanted to destroy every part of the Jew; he therefore used three expressions to denote his real intent. However, the neshama is fed by performance of mitzvos, the ruach is fed by pleasure, and the guf is fed by food. The Vilna Gaon writes that this is the reason for the three mitzvos of Purim; publicly reading the Megillah (see Esther 8:26) feeds the neshama, having joy (see Esther 8:17) feeds the ruach, and feasting (ibid.) literally feeds the guf. There is also an extra mitzvah of giving gifts (ibid. 19). Perhaps Haman’s desire to plunder the Jews’ wealth is the reason we use our resources for this last mitzvah.
R’ Dovid Chadida writes similarly that by using the word “l’hashmid” (“to destroy”), Haman intended to kill the Jews spiritually, by using the word “l’harog” (“to kill”), Haman intended to kill the Jews’ physically, and by using the word “l’aveid” (“ to annihilate”), Haman intended to kill the Jews’ financially. In the verses of Shema (Devarim 6:5), these are the very three things with which Jews are expected to dedicate themselves to love H-Shem.