Shemini in Hebrew "The Eighth" is referred to the level of Binah in Kabbalah. This is where all creation is "fed" and sustained. We have the Eighth day of holiday in Sukkoth, which is after Rosh Hashanah and eighth day of Pesach, which is after Rosh Hashanah (for the months) and they are exactly six months apart. After Pesach we count the Omer for seven weeks and on the first day after that, the fiftieth day and the first day of the eighth week we receive the gift of the Torah on Mount Sinai. For a Mitzvah, a couple is connected after seven days of purity of the woman. The Brit of a new born is made on the eighth day.
Those few examples tell us about the importance of the Eighth level and the ability to connect to Binah.
Moshe was serving in the Holy Tabernacle for seven days and on the eighth day (that was the first day of Nissan and the first day of the year) he was ordered to transfer the duties to Aharon. That was the day of the connection to Binah.
Nadav and Avihu were serving in the process of cleansing (sacrifices) Aharon and the nation from impurities and to be ready for the revelation of light of the Holy tabernacle.
They saw the fire that came from G-d to burn the sacrifices ( 9:24)
"וַתֵּצֵא אֵשׁ, מִלִּפְנֵי יְהוָה, וַתֹּאכַל עַל-הַמִּזְבֵּחַ, אֶת-הָעֹלָה וְאֶת-הַחֲלָבִים"
That process revealed a channel to Binah and the nation saw it and fell to the ground with joy and singing. (9:24) "; וַיַּרְא כָּל-הָעָם וַיָּרֹנּוּ, וַיִּפְּלוּ עַל-פְּנֵיהֶם".
Nadav and Avihu "נָדָב וַאֲבִיהוּא" were one soul. Their name sums to the value of 91 ((Nadav=56)+ (Avihu=24)+(letters=9) + (2 for the words), which according to Kabbalah, tells a lot about their energy. 91 is the value of Amen "אמן " and the combination of "יקוק" the upper and unpronounced name and the lower connection of "אדני". When we answer Amen to a blessing we meditate on "יאקדונקי" (the letter ק replaces the letter ה for obvious reasons). The word for Angel "מלאך" is also 91. All their essences were to serve and draw the light to the world. When they saw the great revelation of Light (verse 9:24) they didn\’t hesitate to act on it as we read on the next verse that also starts a new chapter (10:1) and scarify themselves for the benefit of all. We are actually benefiting from their selfless action until Mashiach comes (B"H Today, Amen).
We can learn about that when Moshe came to Aharon and said (10:3)
"וַיֹּאמֶר מֹשֶׁה אֶל-אַהֲרֹן, הוּא אֲשֶׁר-דִּבֶּר יְהוָה לֵאמֹר
בִּקְרֹבַי אֶקָּדֵשׁ, וְעַל-פְּנֵי כָל-הָעָם, אֶכָּבֵד; וַיִּדֹּם, אַהֲרֹן.".
Rashi explains that with this verse Moshe tells Aharon that the action of Nadav and Avihu put them higher than Moshe and Aharon. On the surface it looks like they did a sin but underneath lays a big secret of spiritual connection.
Moshe\’s word to Aharon reveals some of it. "בִּקְרֹבַי אֶקָּדֵשׁ", literal meaning is "with the ones close to me I will be sanctified". The deeper meaning revealed when looking at the words Moshe used. "בִּקְרֹבַי" comes from the root "קרב " which is also root for "קרבן" ("scarifies") and the word "אֶקָּדֵשׁ" from the root "קדש" that means Holy This word relates to connection to the higher level. "קדיש" that we say during the prayers connects us between the different levels of the prayer connection. "קידוש" that is done after the prayer, makes the final and most important connection of spiritual to physical. The letter "ק" is the letter of the ground " קרקע" the "ד" is the letter for humility "דלות" and "ש" is the letter of completion "שלם" and "שלום" (peace). The "ש" related to fire and built from three letters of "ו" that represent the three lines of Chessed, Gevurah and Tiferet (Avraham Yitzachak and Yaakov). (" חגת"). In summary to connect to the Holy we need to create actions below with selfless and humility in order to connect to the upper. "קידוש־השם" is the higher connection.