According to the Vilna Gaon, Esther requested that Haman’s sons be hanged to make it clear that the Jews were acting in accordance with the will of the king, avoiding any future persecution. By hanging Haman’s sons, it was sign to everyone that the king approved of the Jews’ actions.
Ohel Moshe writes that the people could have theoretically thought that Haman was hanged for attempting to kill Mordechai, the rescuer of the king. Esther wanted it to be very clear that, in actual fact, for generations that this was not some political soap opera, but rather H-Shem did all of this for the sake of the Jews.
R’ Yehonason Eibshutz similarly demonstrates that it is not from Achashverosh, but from H-Shem.
Interestingly, Yalkut Pisron Torah (273) writes that this group of sons were handed over to the Jews in the merit of the Jews’ keeping the mitzva (Devarim 22:6-7) of shiluach hakan (“chasing away the mother bird”).
In the Parsha of Titzaveh, which is usually read before Purim, in the first verse (Shemos 27:20), H-Shem commands the Jewish people to make the clothing of the kohanim using the words, “es bnei Yisroel v’yik’chu.” Rabbi Yosef Freedman points out that the last letters of those four words can be rearranged to spell talui (“hanging”) and the first letters of the same words can be rearranged to spell av v’yud (“the father and ten”).
R’ Moshe Dovid Valle notes that the ten sons of Haman, and Haman himself, hang on the tree together, and those eleven people parallel the eleven1 curses mentioned in the Torah (Devarim 27:15-26) reserved for those who do not keep H-Shem’s Law. Their hanging should remove from us these curses.
Rabbi Yaakov Asher Sinclair (https://ohr.edu/purim/deeper_insights/3440) writes that these dead bodies needed to be hanged because the Talmud (Sanhedrin 97b) promises that Moshiach will come to the Jews even if they do not deserve him. This will occur after a wave of teshuva (“repentance”) takes us over after the evil decrees of a tyrant worse than Haman, himself.
together, there are twelve curses in those verses, not eleven. See
Rashi there (Devarim 27:26) that the twelfth and final of these
curses is a general one that encompasses the entire Torah. Perhaps
this is a reason for R’ Moshe Dovid Valle to have not included it in
his calculation of the number of curses.
The Maamar Mordechai writes that Esther mentions the Jews of Shushan specifically because they had been under the threat of annihilation the earliest, and had knowingly been suffering under the shadow of death all of this time. It was only fair for them to reap the benefits of the victory first.
As understood by Rabbi Jonathan Taub, Malbim notes that this is first time for the remainder of Megillas Esther that Shushan is not called Shushan HaBirah (which he translates as the palace compound). He answers that Esther wanted permission to kill in the city as they had in the palace.
In a very different perspective, R’ Shimon Schwab writes that the verse mentions Shushan specifically because the Jews only killed there because they were afraid elsewhere. Esther wanted to show that faith is the thing that helps them. When the Jews do their part, H-Shem will do His part. He continues that this lack of faith is the reason for a Shushan Purim – it is a sign that the Jews in Shushan did not deserve to join the Jews in celebration on the same day.
The Ben Ish Chai points out that Shushan (300+6+300+50=656) has the same gematria (and even same letters) as sason (“joy”) (300+300+6+50=656), which is what the Jews experienced (Esther 8:16) upon their miraculous rescue from obliteration.
Furthermore, it is also the same gematria as lashon ra (“evil speech”) (30+300+6+50+200+70=656). Evil speech is what Haman tried to use (Esther 3:8) to defame the Jews before Achashverosh when requesting permission to exterminate them.