Rashi explains that Achashverosh was using his recent hanging of Haman to demonstrate his fealty to Esther’s position, giving implicit permission for her to write a new decree. The Persian people will, after all, realize that Esther has the kingdom’s full support.
According to the Ibn Ezra, Achashverosh felt that his people would think Haman’s hanging implied that the earlier decree (written less than a week earlier) was a fraud.
However, the Alshich notes that only the residents of Shushan know Haman was hanged; those living in the remaining cities and villages of the kingdom’s 127 provinces did not know Achashverosh’s feelings on the matter. Therefore, new decrees needed to be written to keep them abreast of the changing political climate.
The Darchei Dovid explains Achashverosh’s reference to Haman’s hanging by quoting the Talmud (Taanis 29a) that a rule in Rome – and presumably in other ancient civilizations – was that when an officer of the court died, all decrees were annulled. This was due to the fact that people considered the death to be a punishment for a seemingly unfavorable decree. Therefore, Achashverosh argues, Haman’s death should have annulled the decree against the Jews.
R’ Elisha Gallico writes that Achashverosh felt people needed to know that Haman’s property was given to Esther (Esther 8:1) because Haman bought the rights to the Jews from Achashverosh (Esther 3:9). His hanging, and the transfer of his property to Esther, effectively bestowing upon Esther control of her people’s fate.
The Dena Pishra explains that Achashverosh was telling Esther to not worry about his people harming the Jews because Haman was not only hanged, but even remained hanging.