Before we can understand what happened on Purim we need to understand that when G-d is not revealed we have nature. Nature hides revealed G-dliness. 
Because Jews have a neshomah which is part of G-d Himself we have an awesome ability; to reach through the curtain of nature to the level where there is revealed G-dliness.
Revealed G-dliness supersedes nature.
An example of this level brought down into the physical world were the miraculous Luchos (the tablets with the Ten Commandments) which Moshe Rabbeinu brought down from Mount Sinai.
Unlike the pictures depicting them, the Luchos were a pair of cubes.
They were made by Hashem with the letters engraved through to the other side. There was no front, there was no back. Although the letters were engraved through to the other side, the writing on the rear face was not reversed! 
The ability for a Jew to tune in to the level of revealed G-dliness is best exemplified in people by regarding a Tzaddik.
We know, for example, nature has no dominion over a Tzaddik.
If a Tzaddik directs water to flow uphill, it will flow uphill. In the Talmud, there is a discussion between two Tzaddikim who considered walking through a wall.  Their decision to desist has nothing to do with whether they are able to do so.
The Alter Rebbe, wanting to bless the moon, stopped the boat on which he was imprisoned without physically interfering with it. 
On another occasion when he was being conveyed to prison the day before Shabbos, the Alter Rebbe announced that he did not travel past a certain time the day before Shabbos, and requested his escort to stop. The guard's mirth gave way to fear and respect when the coach wheels kept breaking despite replacement, so that the Alter Rebbe should not have to travel. 
Tzaddikim are different to other Jews in the level or degree to which they are tuned into G-dliness.
If an ordinary person is tuned into G-dliness properly, he too avoids nature's rule over him.
When a Jew serves Hashem so that his whole being is totally dedicated to Hashem and his whole life is service to Him with everything else an inconvenient interlude, then the inconvenient interludes will bear no dominion over his life.
Purim is about this notion.
The story of Purim is well known to most Jews:
The wicked Haman convinced powerful Achashverosh, King of all the provinces in which Jews then lived, to pass a decree that all Jews be wiped out on a certain day.
The day was chosen by way of lottery.
The killing method was very effective; everybody who killed a Jew would be entitled to keep his property.
People responded well to this persuasive argument, and all Jews were to be annihilated on that day. The Jews fasted for three days, prayed and did teshuvah.
Esther, the niece of the Tzaddik of the generation, Mordechai, was the Queen and changed the King's mind so saving the entire nation.
She achieved this by her service to G-d; the result being that the King instead killed Haman and his sons, and Am Yisrael lived on.
The story of Purim is the story of our being saved miraculously.
It was a miracle because the remedy against Haman is totally without logic.
If somebody is going to kill you with a date set six months in advance, a sensible man leaves town or prepares to fight, meanwhile putting his house into a family trust. Additionally he may try to persuade Government Authorities by the lobby process. To go to synagogue and fast for three days and pray to Hashem is an apparently lunatic method of dealing with the threat. Yet this method and only this method worked. Why?
It is a great secret of Torah  that when Jews behave as Jews, they are not limited by nature.
In nature the result was to be killed. It was above nature for the decree to be overturned, with Haman left swinging from a tree.
Behaving as Jews should allow them to reach out and above the limitations of nature and draw down G-d's essence.
This is what we celebrate on Purim.
Extraordinarily not once is Hashem's name mentioned in the Megillah.
It is the only document of Torah where this is so.
Nevertheless our sages tell us that "King" in the Megillah represents Hashem,  Mordechai is the Jewish nation, and Haman is the Yetzer Hora.
The yetzer hora pleads with the King to kill all Israel but the result was in favor of Mordechai and Jews forever.
On Purim the story of Purim, the Megillah, is read in synagogue.
The Rebbe explains in great depth a verse from the Megillah,  "The King woke up". We have said the King is Hashem; can it be said that Hashem sleeps?
When Jews behave according to the will of G-d, He is, as it were, awake; when we behave to the contrary, Hashem is, as it were, asleep.
Hashem asleep, is a lack of G-dly revelation, and so nature is what we see and relate to. Hashem awake means that there is a revelation of G-dliness which we can see and relate to.
When Moshe Rabbeinu received the Tablets on Mount Sinai, when all Jews were unified in Hashem's will, Hashem was therefore, as it were, awake. His present to the Jewish nation, the Ten Commandments, therefore were a level of revealed G-dliness and so the Tablets had no front and no back.
The moment the actions of Jews degenerated, so returning Hashem to sleep, the revelation of G-dliness was removed and the Tablets were broken.
All powers given to human beings come from the soul.
A human soul is equipped with various powers, for example, the power to see and the power to hear. The power exists independently of the limb which is the receptacle of that power.
So if a person has the power of sight, he has the power of sight independently of the organ that sees, independently of the eye. If he were to, notionally, take out his eye, vision would be interrupted because the instrument was missing.
The moment the eye is restored, he will see again. This is so because the power to see remains constant but is only available to be revealed by the presence of that instrument.
Similarly, the power to hear.
If the ear is filled with wax, sound waves may no longer have access to the ear drum and the person will not hear. But, of course, the power to hear remains unimpaired.
Consider however the finest ear without a soul; kill the man but preserve perfect ears. Can he hear? The power is in the soul in a general level and in the ear in a particular level.
A fascinating thing happens to a man's powers when he is asleep.
The powers cannot have disappeared because the person is still alive. The particular powers to the limbs however are reduced, weakened. Secondly, they are mixed up and so it is possible to dream dreams of opposites.
Everybody shares the frustration of dreaming of walking over sky or putting on a book instead of a shirt. Such opposites in dreams co- exist because the powers which co-exist in the soul are in a jumbled up state.
So it is with the soul of the world, Hashem.
When Hashem, as it were, is asleep, the level of revelation of G-dliness in the world is weak and is all jumbled up. Bad co-exists with good - not only does it co-exist with good, it seems to totally overtake it.
We cannot walk the streets at night without the risk of some head- shaven demented savage trying to attack and kill us.
That night, says the verse, the King woke up; the King was no longer asleep. There was again a revelation of G-dliness. The Jews prayed and fasted for three days and so the King woke up. Esther, one of the most beautiful women in history,  was to seek to have King Achashverosh reverse Haman's decree and save the Jewish nation.
A natural method would involve the bath house, perfume, the hairdresser and perhaps the lingerie shop. Instead, Esther took the non-natural course of fasting for three days.
Skin goes gray after three days fasting and is accompanied by foul smelling breath. In the moment of her least appeal, Esther is welcomed by the King. This result is outside nature because she was functioning outside nature, the level a Jew can function if he lives as a Jew.
With halitosis and gray skin, Esther looked wonderful to Achashverosh and he listened to her.
The Jews in the provinces, instead of appointing a committee to lobby and a sub-committee to fundraise, went to synagogue.
They fasted for three days and they davened to Hashem and were thus able to connect to the inner aspect of G-dliness which is above nature.
This is what we celebrate on Purim. Nature is totally eclipsed and every single Jew is saved.
It is fascinating that, in those moments when there shines holiness revealed by Jews living as Jews, Haman cannot exist. He must die.
This is what Moshiach (the Messianic Redeemer) is.
To have Moshiach is to eradicate Haman.
There will then be a revelation of G-dliness so strong that it will be impossible for evil to exist. This revelation of G-dliness can only come from Jews living and learning together as Jews.
1. See for example Likkutei Sichos, Vol. 20, p. 36.
2. See Talmud Yerushalmi Shekalim 6:1.
4. See Sefer HaToldos of the Alter Rebbe
5. See Beis Rebbi, Chapter 21.
6. See Likkutei Sichos, Vol. 1, p. 213ff.
7. See Yalkut Shimoni on the Megillah 6:1.
8. For the following explanation see Sefer HaMaamarim Melukot Vol. 2, p. 265ff. (See also Sefer HaMaamarim 5710, p. 1ff).
9. See Megillas Esther 2:17 and Talmud Megillah 13a.