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.