The Gemara states as follows:
אסתר מן התורה מנין? שנאמר "ואנכי הסתר אסתיר פני."
What is the source for "Esther" from the Torah? From the verse, "I will hide (astir) My presence on that day."
From here we see that the concealment of Hashem's presence is the central theme of the holiday of Purim. Pesach, Shavuos and Sukkos commemorate the openly revealed miracles experienced by our forefathers: the Exodus and the splitting of the Red Sea; the momentous event of Kabbalas HaTorah; and the Clouds of Glory that protected them in their travels through the desert. Chanukah also celebrates a supernatural, miraculous event: the jar that held enough oil to burn for only one night but miraculously burned for eight.
These were all miracles that transcended the boundaries of nature. They clearly demonstrated Hashem's ultimate mastery over all creation and His direct interaction in our affairs.
Purim, in contrast, commemorates no openly revealed miracle. Our deliverance from Haman's plot was a natural series of events, in an episode of palace intrigue that could have occurred in any royal court. Here, Hashem did not override the laws of nature. He worked within the framework of nature to orchestrate our redemption in a way that seemed perfectly natural.
A revealed miracle inspires awe and devotion in the hearts of all who see it. The miracle of Purim, on the other hand, cannot even be seen unless one already has the sense of devotion necessary to peer behind the veils that mask Hashem's presence.
Of miracles such as these, the Torah foretells, "I will hide My presence on that day." In context, this verse refers to the troubles and travails that befall us in Golus:
וַיֹּאמֶר ה' אֶל מֹשֶׁה הִנְּךָ שֹׁכֵב עִם אֲבֹתֶיךָ וְקָם הָעָם הַזֶּה וְזָנָה אַחֲרֵי אֱלֹהֵי נֵכַר הָאָרֶץ אֲשֶׁר הוּא בָא שָׁמָּה בְּקִרְבּוֹ וַעֲזָבַנִי וְהֵפֵר אֶת בְּרִיתִי אֲשֶׁר כָּרַתִּי אִתּוֹ. וְחָרָה אַפִּי בוֹ בַיּוֹם הַהוּא וַעֲזַבְתִּים וְהִסְתַּרְתִּי פָנַי מֵהֶם וְהָיָה לֶאֱכֹל וּמְצָאֻהוּ רָעוֹת רַבּוֹת וְצָרוֹת וְאָמַר בַּיּוֹם הַהוּא הֲלֹא עַל כִּי אֵין אֱלֹהַי בְּקִרְבִּי מְצָאוּנִי הָרָעוֹת הָאֵלֶּה. וְאָנֹכִי הַסְתֵּר אַסְתִּיר פָּנַי בַּיּוֹם הַהוּא עַל כָּל הָרָעָה אֲשֶׁר עָשָׂה כִּי פָנָה אֶל אֱלֹהִים אֲחֵרִים.
Hashem said to Moshe, "You will lie with your forefathers, and this nation will rise up and stray after the foreign deities of the land to which they come. They will forsake Me and break the covenant that I made with them. Then My anger will flare against them on that day. I will forsake them and hide My presence from them. They will be consumed, and many terrible misfortunes will befall them. They will say on that day, 'Since Hashem is not in our midst, all these troubles have befallen us.' I will hide My presence from them on that day, because of all the evil they have done, since they turned towards other gods.
The Gemara interprets this verse as a reference to the miracle of Purim, which occurred in Golus. It is a paradigm for the manner by which Hashem protects us even in the darkest and harshest moments of Golus.
Today there are no pillars of fire to lead us, no clouds of glory to surround us, and no thunderous voice calling to us from the Heavens. In the hardships that befall Klal Yisrael in Golus, we can hardly draw courage from the overt miracles that our forefathers experienced, since they are unavailable to us. Instead, we must draw courage from the miracle of Purim, which showed how Hashem's protective hand is still by our side, even as His presence is hidden from our eyes.
Without overriding the forces of nature, Hashem subtly twists them to our benefit. He guides the political stratagems of nations, stands behind all thrones, and directs the course of history for the sake of His beloved children. A Persian monarch executes his wife for her insolence; a Jewish woman is taken to the palace against her will; an assassination attempt is foiled. When viewed individually, they all seem like normal events, but when viewed together in hindsight they are revealed as a miracle no less wondrous than the splitting of the Red Sea.
"Hashem's Plan Shall Prevail"
The Midrash states as follows:
מהו דכתיב רבות מחשבות בלב איש ועצת ה' היא תקום, רבות מחשבות בלב איש זה המן הרשע שהיה מחשב בלבו ואומר למחר אני משכים ואומר למלך מעשיו של מרדכי והוא משליחו לידי, והוא לא היה יודע שמחשבתו של מקום קודמת לשלו דכתיב ועצת ה' היא תקום.
"Many schemes are in the heart of man, but Hashem's plan shall prevail." Many schemes were in the heart of Haman, who thought to himself, 'Tomorrow I will rise and tell the king of Mordechai's deeds, and he will deliver Mordechai into my hands.' He did not realize that Hashem's plans foresaw his own, as it is written, 'Hashem's plan shall prevail.'"
Rebbe Simcha Bunim of Peshischa explained that it is no wonder that Hashem's plans prevailed over the schemes of Haman. The Midrash would not even take note of this. Rather, the Midrash means to say that Hashem's plans prevailed by making use of Haman's own schemes against him. The gallows he built for Mordechai were used to hang him and his ten sons, and the date he set for the annihilation of the Jews marked our greatest victory.
אֲשֶׁר שִׂבְּרוּ אֹיְבֵי הַיְּהוּדִים לִשְׁלוֹט בָּהֶם וְנַהֲפוֹךְ הוּא אֲשֶׁר יִשְׁלְטוּ הַיְּהוּדִים הֵמָּה בְּשֹׂנְאֵיהֶם.
The enemies of the Jews thought to overpower them, but it was turned about, and the Jews overpowered their enemies.
The Shem M'Shmuel adds that the eradication of Amalek must precede the building of the Beis HaMikdash. Since there was then no Jewish king to lead the war against Amalek, Hashem instead brought about a course of events in which Haman, the scion of King Agag of Amalek, sealed his own doom. This set the stage for the construction of the second Beis HaMikdash, which could have been built in no other way.
Haman thought that Adar was an ill-omened month for Bnei Yisrael, since it marked the passing of our leader Moshe Rabbeinu. Little did he realize that Adar is in fact an auspicious month for us, since it also marks Moshe's birth. In this month, the worst of our tragedies turn out to be our greatest fortune. Hashem's harsh judgment was revealed as but a mask for His love and mercy.
Threads in the Tapestry
Rather than rolling up the left side of the Megillah as it is read, the left side is folded over and over,  so that when Megillah is finally completed the entire story is exposed at once. Each incident of the Megillah presents no proof of Hashem's involvement. Only when the entire story is seen at once from beginning to end, does every incident reveal itself as a thread in a tapestry that clearly displays Hashem's miraculous intervention.
Esther made a special request that the Sanhedrin establish Purim as a holiday in which Megillas Esther is read, in order that the memory of the miracles of Purim might never be forgotten. She realized that this holiday would be of paramount importance to support and encourage Klal Yisrael in the darkest epochs of our Golus. Only by contemplating the miracle of Purim, can we perceive how Hashem orchestrates all the events that befall us to our ultimate benefit.
The Arizal notes that although Hashem's Name is never explicitly mentioned throughout the Megillah, it does appear as an acronym on several occasions. The first letters of יבא המלך והמן היום("Let the king and Haman come today") and the last letters of כי כלתה אליו הרעה ("The harm against [Haman] was final ") spell Hashem's Name.
The verse with Hashem's Name in its first letters depicts Haman's rise to power and the ominous threat of destruction that loomed over Klal Yisrael. The second verse, in which Hashem's Name appears in the last letters, depicts Haman's downfall and our salvation from his nefarious plan.
This highlights the central lesson of the Megillah. Hashem's Name appears at the beginning to spell the words of destruction, yet at the end it is revealed that His Name really spells the words of salvation. The very threats that at first seem so menacing are ultimately revealed as the very tools by which Hashem redeems us. ממכה עצמה מכין רטיה - "From the illness itself, Hashem creates the cure," as we say in Selichos.
Concealment and Revelation
The word "Megillah" hints to revelation (giluy), while the word "Esther" hints to concealment (astir). By contemplating the events of the Megillah, we reveal Hashem's presence, even as He conceals himself behind a mask of nature. This is the depth of the custom to dress up in costumes on Purim. We hide ourselves behind masks, since Hashem also hides His presence even as He brings about our deliverance.
The Gemara explains that there is no need to recite Hallel on Purim, since we fulfill the obligation to recite Hallel by reading the Megillah instead. Why do we suffice with a replacement for Hallel? Is it so difficult to recite the standard Hallel as well? Furthermore, the Megillah does not seem to be a prayer of praise and thanksgiving like Hallel, but a simple story of events.
Yet we can explain that this simple story is the greatest and most fitting form of Hallel for Purim. Since the very essence of the day is to reveal the miracles hidden behind nature, our Hallel is also hidden within the reading of the Megillah. We gaze into the events of the Purim story to discover miracles no less wondrous than the "sea that saw and fled; the Yarden turned back on itself" that we recall in Hallel.
 Devarim 31:18; Chullin 139b
 Midrash Mishlei 19:21
 Esther 9:1
 Shem M'Shmuel, Tetzaveh 5672
 Shulchan Aruch, Orach Chaim 690:17
 Megillah 7a
 Pri Eitz Chaim, Shaar HaPurim ch. 6
 Megillah 14b