The core of the explanation of the concept1 is thatוּנְקֻדַּת הַבֵּאוּר בָּזֶה,
the purpose of the Chanukah lights is to illuminate the darkness.דְּעִנְיָנָם שֶׁל נֵרוֹת חֲנֻכָּה הוּא לְהָאִיר אֶת הַחֹשֶׁךְ.
The last two points – that Chanukah reflects a transcendent light above the structure of the Spiritual Cosmos and that the Chanukah lights are kindled after sunset –are interrelated. The Menorah had seven lamps because the purpose of the Menorah was to illuminate the world as it exists within its natural framework. As explained below, the Greeks sought to prevent that light from shining forth. Indeed, they brought darkness to the world.
The light of the Menorah, i.e., the Divine light appropriate to the spiritual structure of the world, is not sufficient to combat that degree of darkness. On the contrary, the Greeks were able to impart impurity to that light. To overcome the darkness they brought and illuminate it, a higher light – one that transcends the spiritual structure of the world – was necessary.
To present the concept in its historical context:
The Chanukah miracle transpired after the Greeks entered the Sanctuary and imparted impurity to all the oil in the Sanctuary.2דְּהִנֵּה הַנֵּס דַּחֲנֻכָּה הָיָה לְאַחֲרֵי שֶׁנִּכְנְסוּ הַיְּוָנִים לַהֵיכָל וְטִמְּאוּ כָּל הַשְּׁמָנִים שֶׁבַּהֵיכָל,
The intensification of the darkness (brought about by the Greeks)שֶׁהִתְגַּבְּרוּת הַחֹשֶׁךְ (דְּהַיְּוָנִים)
Indeed, our Sages3 specifically associated darkness with the Greeks, interpreting the term “darkness” mentioned at the beginning of the narrative of creation4 as a reference to the sovereignty of Greece.
was so great thatהָיְתָה עַד כְּדֵי כָּךְ,
not only did they enter the Sanctuary and impart impurity to it,דְּלֹא רַק שֶׁנִּכְנְסוּ לַהֵיכָל וְטִמְּאוּהוּ,
they imparted impurity to all the oil in the Sanctuary.אֶלָּא שֶׁטִּמְּאוּ גַּם אֶת כָּל הַשְּׁמָנִים שֶׁבַּהֵיכָל.
As will be explained in sec. 4, the oil represents a higher spiritual rung than the Sanctuary. The Sanctuary is identified with the Sefirah of Binah, while oil is identified with the Sefirah of Chochmah.
Thus, oil is identified with the level of Kodesh, “holiness,” (as it is written:5 “oil, a holy ointment”).6דְּשֶׁמֶן הוּא בְּחִינַת קֹדֶשׁ (כְּמוֹ שֶׁכָּתוּב: "שֶׁמֶן מִשְׁחַת קֹדֶשׁ"),
As the Zohar states:7 “Kodesh, “holiness,” is a distinct quality.”קֹדֶשׁ מִלָּה בְּגַרְמֵיהּ
There are entities that are described as kadosh (קדוש), “holy,” i.e., holiness has been imparted to them. This is alluded to by the vav whose form indicates that light is drawn down from its source. Kodesh (קדש), by contrast, refers to holiness in its essence, above any connection to a specific entity.
(The uniqueness of this spiritual level is reflected in the physical property of oil, for oil floats on top of all other liquids8 and does not mix with them.)9(בְּדֻגְמַת הַשֶּׁמֶן שֶׁצָּף עַל גַּבֵּי כָּל הַמַּשְׁקִין וְאֵינוֹ מִתְעָרֵב עִמָּהֶם).
From this, it can be understood that the fact that the Greeks imparted impurity to all the oil in the Sanctuary as a whole,דְּמִזֶּה מוּבָן, שֶׁזֶּה שֶׁטִּמְּאוּ אֶת כָּל הַשְּׁמָנִים שֶׁבַּהֵיכָל,
is more significant (in this dimension) than the fact that they imparted impurity to the Sanctuaryנוֹגֵעַ (בִּפְרָט זֶה) עוֹד יוֹתֵר מִזֶּה שֶׁטִּמְּאוּ אֶת הַהֵיכָל,
even though the sanctity of the Sanctuary is a lofty level within the ten levels of holiness mentioned by our Sages.10אַף שֶׁקְּדֻשַּׁת הַהֵיכָל הִיא דַּרְגָּא נַעֲלֵית בְּעֶשֶׂר קְדֻשּׁוֹת.
Indeed, in their reckoning, it is only one level less than the Holy of Holies.
The severity of the darkness brought into the world by the Greeks was so great that they were even able to impart impurity to the oil of the Menorah, i.e., to prevent the Menorah’s light from shining into the world with its intrinsic purity. The depth of the descent is reflected in the fact that even after “Your sons came… and purified Your Sanctuary,”11 nevertheless,וְכִדְמוּכָח גַּם מִזֶּה שֶׁגַּם לְאַחֲרֵי שֶׁבָּאוּ בָנֶיךָ כוּ' וְטִהֲרוּ אֶת מִקְדָּשֶׁךָ,
in order to kindle the Menorah with pure oil a miracle – indeed, several miracles – were necessary,הִנֵּה בִּכְדֵי שֶׁיּוּכְלוּ לְהַדְלִיק אֶת הַמְּנוֹרָה הֻצְרְכוּ לְנֵס דַּוְקָא,
beginning with the miracle that they found a cruse of pure oil sealed12 with the seal of the High Priest.מַתְחִיל מֵהַנֵּס שֶׁמָּצְאוּ פַּךְ אֶחָד שֶׁמֶן טָהוֹר חָתוּם בְּחוֹתָמוֹ שֶׁל כֹּהֵן גָּדוֹל,
Why is Chanukah considered an eight-day miracle? The cruse of oil found contained enough oil to burn for one day. Thus, it burned miraculously for only seven days.13 Among the answers given is that the very discovery of the cruse of pure oil is considered a miracle.14
To focus on the spiritual dimension of that oil: The oil with the High Priest’s seal represented
G‑d’s infinite light that transcends [the structure of] the Spiritual Cosmos.אוֹר אֵין סוֹף שֶׁלְּמַעְלָה מֵהִשְׁתַּלְשְׁלוּת,
In relation to this light – to borrow a Scriptural phrase15 – “Even darkness will not shade.”שֶׁלְּגַבֵּי אוֹר זֶה גַּם חֹשֶׁךְ לֹא יַחְשִׁיךְ,
On the contrary, this light illuminates the darkness and causes the darkness itself to shine.וְאַדְּרַבָּא אוֹר זֶה מֵאִיר אֶת הַחֹשֶׁךְ שֶׁגַּם הַחֹשֶׁךְ יָאִיר,
Implied is that not only is this light able to penetrate the darkness, but that it has the potential to transform the darkness itself into light. It is written,16 “...like the superiority of light over darkness.” Chassidus17 explains that the phrase translated as “over darkness” actually reads min hachoshech, “from the darkness.” A superior light is produced by the transformation of darkness into light.
As mentioned above, sparks of G‑dliness exist within every entity in this world. These sparks reflect varying spiritual levels. In a unique process of causation, the loftier the spark’s source, the further it descends. Hence, sparks whose source is in the loftiest rungs of G‑dliness are found in the realm of evil.
Man’s Divine service involves breaking through the negative character produced by this descent and revealing the inner G‑dly potential that lies within, transforming darkness into light. Similarly, the Divine potential alluded to by the cruse of oil sealed with the seal of the High Priest has within its power the potential to transform the darkness of the Greeks into light.
This transformation is alluded to by the phrase,18 “And G‑d will illuminate my darkness.”19 Chassidus explains that the vav added to G‑d’s name alludes to additional light. A vav alludes to drawing down G‑dly energy from a higher source. Essential G‑dly light is drawn down to G‑d’s name Havayah and that generates the potential to illuminate darkness.וַהֲוָיָ׳ (בְּתוֹסֶפֶת וָא"ו) יַגִּיהַּ חָשְׁכִּי.
On this basis, we can understand why the mitzvah of kindling Chanukah lights begins specifically after sunset:וְזֶהוּ שֶׁנֵּר חֲנֻכָּה מִצְוָתָהּ מִשֶּׁתִּשְׁקַע הַחַמָּה דַּוְקָא,
Because the purpose of kindling Chanukah lights is to illuminate darkness.כִּי עִנְיַן נֵרוֹת חֲנֻכָּה הוּא לְהָאִיר אֶת הַחֹשֶׁךְ.
This also explains why there are eight Chanukah lights although there were only seven lights to the Menorahוְזֶהוּ גַּם מַה שֶּׁנֵּרוֹת חֲנֻכָּה הֵם שְׁמוֹנָה דַּוְקָא
(and similarly, why there are eight days of Chanukah).20(וְכֵן יוֹמֵי דַּחֲנֻכָּה תְּמָנְיָא אִינוּן),
For, (as explained in sec. 1,) the number eight reflects a transcendent potential, above the structures of the Spiritual Cosmos.שֶׁמִּסְפַּר שְׁמוֹנֶה מוֹרֶה (כַּנַּ"ל ס"א) עַל בְּחִינָה שֶׁלְּמַעְלָה מֵהִשְׁתַּלְשְׁלוּת,
For specifically this quality of transcendent light makes it possible even to illumine darkness.שֶׁמִּצַּד בְּחִינָה זוֹ דַּוְקָא אֶפְשָׁר לְהָאִיר גַּם אֶת הַחֹשֶׁךְ.

Summary

The purpose of the Chanukah lights is to illuminate the darkness. The Greeks engendered a climate of spiritual darkness, as reflected in the fact that they “imparted impurity to all the oil in the Sanctuary.” The Jews discovered a cruse of oil sealed with the seal of the High Priest, which enabled them to combat and overcome this darkness. In commemoration of this miracle, we kindle the Chanukah lights after nightfall, illuminating the darkness of the night.
On this basis, we can understand why eight Chanukah lights are kindled. As mentioned above, eight reflects a transcendent light above the structures of the Spiritual Cosmos. Tapping this light generates the potential to illuminate spiritual darkness.