On a simple level, the Ibn Ezra, Rambam (Mishneh Torah Hilchos Taanios 5:5), and Me’am Loez write that the fast to which this verse refers is the fast of Esther. However, according to the Talmud (Megilla 16b), these words are meant to be read with the beginning of the next verse. Therefore, it was through both fasting and Esther’s words that the Jews earned the merit to be rescued from total annihilation.
The M’nos HaLevi writes that this means that, just as the Jews accepted upon themselves the fasts of the prophets and fasting for the Temple’s destruction, they accepted Purim with all of its rules.
As Malbim explains, Esther and Mordechai used the prophet’s (Zecharya 8:19) establishing other fasts as proof that holidays can be established without violating the Torah’s (Devorim 4:2, see Ramban there) prohibition against bal tosif (“adding to the Torah’s existing laws”).
The Ginzei HaMelech notes the correspondence between Purim and fast days. He relates it to Yalta’s saying in the Talmud (Chulin 109b) that the Torah permits everything it forbids. In other words, the joy of Purim counterbalances the sadness of the fast days, zeh l’umas zeh.
This fits well with the Ksav Sofer noting the Talmud’s (Taanis 29a) parallel between the months of Adar and Av; just as mishenichnas Adar, marbin b’simcha (“when Adar begins, we increase our joy”), so too mishenichnas Av, mimaatin b’simcha (“when Av begins, we decrease our joy”).
The Sfas Emes notes a similar parallel between Purim and Yom Kippur by applying the words of the wisest of all men (Mishlei 18:21) that maves v’chaim b’yad halashon (“death and life are controlled by the tongue”). In other words, H-Shem’s judging the Jews occurs on both days, and is manifest in how we utilize our power of speech to maintain peace and unity.
Furthermore, the Maharal adds that we would logically assume Purim should be a time for fasting, considering the reasons H-Shem had for annihilating us. Instead we customarily drink ad d’lo yada to sublimate our logic in order need to recognize that our salvation does not come from our effort, but from H-Shem’s help.
Either way, fasting led to the Purim miracle, so R’ Moshe Dovid Valle notes that the word hatzomos (“the fasts”) is written in plural because an individual may choose to fast all three days of Purim (Taanis Esther, Purim, and Shushan Purim), but this is not for the masses.