After the mitzvah of dwelling in the sukkah and the mitzvah of taking the four speciesוְהִנֵּה לְאַחֲרֵי מִצְוַת יְשִׁיבָה בַּסֻּכָּה וּמִצְוַת נְטִילַת ד' מִינִים
comes the mitzvah of offering the festival sacrifices,בָּאָה הַמִּצְוָה דְּהַקְרָבַת קָרְבְּנוֹת הֶחָג
The term “after” is used because of the chronological sequence of these three mitzvos: the mitzvah to dwell in the sukkah begins at night, at the onset of the festival; the taking of the four species may be performed immediately in the morning; and offering the musaf sacrifices associated with the festival was not carried out until many other phases of the sacrificial worship of the day were performed.
It is possible to say that the chronological sequence also reflects a spiritual progression. First, it is necessary to elicit the essential unity that transcends all distinctions; (this is accomplished through dwelling in the sukkah). Afterwards, this unity is drawn down into the realm where particular distinctions exist, merging these distinct entities in oneness (the four species). Subsequently, this unity is drawn down into the world at large, the 70 nations (offering the 70 bulls).
the fundamental Sukkos festival sacrifices being the 70 bulls,שֶׁעִקָּרָם הֵם הָע' פָּרִים
[they were given that distinction because the bull (the ox) is “king of the domesticated animals”],1[כֵּיוָן שֶׁפָּר (שׁוֹר) הוּא הַמֶּלֶךְ שֶׁבַּבְּהֵמוֹת]
which correspond to the 70 nations of the world.2כְּנֶגֶד שִׁבְעִים אֻמּוֹת.
Through offering the 70 bulls –דְּעַל יְדֵי הַקְרָבַת הָע' פָּרִים
[and similarly, at present, through reciting the verses describing the offering of the 70 bulls,[וְעַל דֶּרֶךְ זֶה עַכְשָׁיו עַל יְדֵי אֲמִירַת פְּסוּקֵי הַקָּרְבָּנוֹת דְּע' פָּרִים,
in the spirit of the verse “We will compensate for the sacrifice of bulls through our lips”]3 –וּנְשַׁלְּמָה פָרִים שְׂפָתֵינוּ]
the refinement of the nations of the world is accomplished.4נַעֲשֶׂה בֵּרוּר הָאֻמּוֹת.
It is possible to explain that the connection between the refinement of the nations through the offering of the seventy bulls and the mitzvah of taking the four species isוְיֵשׁ לוֹמַר, שֶׁהַשַּׁיָּכוּת דְּבֵרוּר הָאֻמּוֹת עַל יְדֵי הַקְרָבַת הָע' פָּרִים לְמִצְוַת נְטִילַת ד' מִינִים הִיא,
that the ultimate perfection of the victory of the Jewish people, which is revealed through taking the four species –כִּי תַכְלִית שְׁלֵמוּת הַנִּצָּחוֹן דְּיִשְׂרָאֵל שֶׁמִּתְגַּלֶּה עַל יְדֵי נְטִילַת ד' מִינִים
[as explained in sec. 1 above based on the statement from the Midrash, by the Jews taking their lulavim and esrogim in hand it is made known that the Jews prevailed in the judgment of Rosh HaShanah] –[דְּעַל יְדֵי שֶׁיִּשְׂרָאֵל לוּלְבֵיהֶן וְאֶתְרוֹגֵיהֶן בְּיָדָן יוֹדְעִים דְּיִשְׂרָאֵל אִינוּן נִצּוּחַיָּא, כַּנִּזְכָּר לְעֵיל מִמִּדְרָשׁ]
הוּא, שֶׁגַּם הָאֻמּוֹת יַסְכִּימוּ לָזֶה,
comes about when even the non-Jewish nations themselves concur with and accept the judgment.הוּא, שֶׁגַּם הָאֻמּוֹת יַסְכִּימוּ לָזֶה,
When a nation overcomes an enemy and defeats it in battle, hatred and enmity persist between the two. It is very likely that hostilities will resume as soon as the vanquished nation regains its strength. When, by contrast, the enemy accepts the victor and seeks to establish peace with it, the triumph is complete and enduring.
Similarly, through the offering of the 70 bulls,וְעַל יְדֵי הַקְרָבַת הָע' פָּרִים
which correspond to the 70 nations, they are refined and perfected,כְּנֶגֶד ע' אֻמּוֹת הֵם מִתְבָּרְרִים וּמִזְדַּכְּכִים
to the extent that they accept the victory of the Jewish people.וְעַד שֶׁהֵם מַסְכִּימִים לְהַנִּצָּחוֹן דְּיִשְׂרָאֵל.
On this basis, we can appreciate the connection between the recitation of Hallel during the seven days of Sukkos with the offering of the 70 bulls.וְזוֹהִי גַם הַשַּׁיָּכוּת דַּאֲמִירַת הַלֵּל בְּז' יְמֵי הַסֻּכּוֹת לְהַקְרָבַת הָע' פָּרִים
[For, as our Sages explain,5 the reason that the entire Hallel is recited on all the days of the holiday of Sukkos is because a different number of sacrifices were offered on each of the seven days.]6[דְּזֶה שֶׁגּוֹמְרִין אֶת הַהַלֵּל בְּכָל יְמֵי הֶחָג הוּא לְפִי שֶׁהֵם חֲלוּקִים בְּקָרְבְּנוֹתֵיהֶם],
For the refinement and perfection of the nations that is achieved via the offering of the 70 bulls leads to:כִּי עַל יְדֵי הַבֵּרוּר וְהַזִּכּוּךְ שֶׁנַּעֲשֶׂה בְּהָאֻמּוֹת עַל יְדֵי הַקְרָבַת הָע' פָּרִים, נַעֲשֶׂה
“Praise G‑d, all you nations; laud Him, all you peoples. For His kindness has overwhelmed us.”7הַלְלוּ אֶת הֲוָיָ׳ כָּל גּוֹיִם שַׁבְּחוּהוּ כָּל הָאֻמִּים כִּי גָבַר עָלֵינוּ חַסְדּוֹ גו',
The nations and peoples praise G‑d for showing us overwhelming kindness. They thank G‑d for His kindness to the Jews because they recognize the virtues of the Jewish people.שֶׁהַגּוֹיִם וְהָאֻמִּים מְשַׁבְּחִים אֶת הֲוָיָ׳ עַל זֶה שֶׁגָּבַר עָלֵינוּ חַסְדּוֹ.

Summary

The mitzvos of the holiday of Sukkos reflect a spiritual sequence: First, the essential unity that transcends all distinctions is elicited through the sukkah. Afterwards, this unity is drawn down into the realm in which particular distinctions exist through the four species. Subsequently, this unity is in turn drawn down into the world at large, the 70 nations, via the offering the 70 bulls.
These sacrifices refine and elevate the nations to the extent that they accept G‑d’s judgment that the Jews will prevail. They praise Him “because His kindness has overwhelmed us,” i.e., they offer thanks to Him for the kindness He showers on the Jewish people.