כְּגַוְונָא

Origins: Though how it got added to the Chassidishe Nusach for Kabbalas Shabbos remains ambiguous, it was likely introduced by the early chassidim of 18th century Brody, Ukraine. Certainly, though, by the third generation of Chassidus Kagavna had become an accepted piyut said by chassidic communities all over.

DISCLAIMER- I am not a Mekubbal. This explanation of Kagavna is strictly focused on what I have found about this piyut in terms of why I believe it's worth saying. I will be leaving out any explanations regarding Kabbalistic Messorah.

ספר הזהר, תרומה י״ד

כְּגַוְונָא דְּאִינּוּן מִתְיַיחֲדִין לְעֵילָּא בְּאֶחָד, אוּף הָכִי אִיהִי, אִתְיַחֲדַת לְתַתָּא בְּרָזָא דְּאֶחָד, לְמֶהוֵי עִמְּהוֹן לְעֵילָּא חַד לָקֳבֵל חַד, קוּדְשָׁא בְּרִיךְ הוּא אֶחָד לְעֵילָּא, לָא יָתִיב עַל כּוּרְסְיָיא דִּיקָרֵיהּ, עַד דְּאִיהִי אִתְעֲבִידַת בְּרָזָא דְּאֶחָד כְּגַוְונָא דִּילֵיהּ, לְמֶהֱוֵי אֶחָד בְּאֶחָד[1]. וְהָא (קל''ד ע''א) אוֹקִימְנָא רָזָא דַּיְיָ' אֶחָד וּשְׁמוֹ אֶחָד. רָזָא דְּשַׁבָּת, אִיהִי שַׁבָּת, דְּאִתְאַחֲדָא בְּרָזָא דְּאֶחָ''ד,

Zohar, Terumah 14

Just as [or: When] they are united above, so does She unite below in the mystery of One, to be with them above, One corresponding to One. The blessed Holy One, one above, does not sit upon His Throne of Glory until She becomes, by mystery of One, like Him, to be one with one.[1] Mystery of Sabbath: She is Sabbath—united in the mystery of one,

ספר הזהר, תרומה י״ד

לְמִשְׁרֵי עֲלָהּ רָזָא דְּאֶחָד. צְלוֹתָא דְּמַעֲלֵי שַׁבְּתָא, דְּהָא אִתְאַחֲדַת כּוּרְסְיָיא יַקִּירָא קַדִּישָׁא, בְּרָזָא דְּאֶחָ''ד, וְאִתְתַּקָּנַת לְמִשְׁרֵי עֲלָהּ מַלְכָּא קַדִּישָׁא עִלָּאָה. כַּד עַיֵיל שַׁבְּתָא, אִיהִי אִתְיַחֲדַת וְאִתְפַּרְשַׁת מִסִּטְרָא אַחֲרָא, וְכָל דִּינִין מִתְעַבְּרִין מִינָהּ, וְאִיהִי אִשְׁתְּאָרַת בְּיִחוּדָא דִּנְהִירוּ קַדִּישָׁא, וְאִתְעַטְּרַת בְּכַמָּה עִטְרִין לְגַבֵּי מַלְכָּא קַדִּישָׁא, וְכָל שׁוּלְטָנִי רוּגְזִין וּמָארֵי דְּדִינָא כֻּלְּהוּ עַרְקִין, (נ''א ואתעברו מינה) וְלֵית שׁוּלְטָנוּ אַחֲרָא בְּכֻלְּהוּ עָלְמִין[2]. וְאַנְפָּהָא נְהִירִין בִּנְהִירוּ עִלָּאָה, וְאִתְעַטְּרַת לְתַתָּא בְּעַמָּא קַדִּישָׁא, וְכֻלְּהוּ מִתְעַטְּרִן בְּנִשְׁמָתִין חַדְתִּין[3]. כְּדֵין שֵׁירוּתָא דִּצְלוֹתָא, לְבָרְכָא לָהּ בְּחֶדְוָה, בִּנְהִירוּ דְּאַנְפִּין, וְלוֹמַר בָּרְכוּ אֶת ה' הַמְבוֹרָךְ.

Zohar, Terumah 14

so that mystery of One may settle upon Her. Prayer for the entrance of Sabbath: then the Holy Throne is united in mystery of One, arrayed for the supernal Holy King to rest upon Her. When Sabbath enters She unites, and separates [herself] from the Other Side, all judgments removed from Her. And She remains unified in holy radiance, adorned with many crowns for the Holy King. All powers of wrath and masters of judgment all flee (and pass away from her) and no alien power reigns in all the worlds.[2] Her face shines with supernal radiance, and She is adorned below by the Holy People, all of whom are adorned with new [or: joyous] souls.[3] Then, beginning of prayer, blessing Her with joy and beaming faces, saying: “Bless Hashem, the blessed one !”

"The Shabbat as [a] time of unity is also marked by [the] cessation of all the inner conflicts and tensions within the universe. All powers of wrath and masters of judgment flee and no alien power reigns in the world. The world, which is ordinarily a field of conflict between the powers of love and graciousness and those of wrath, is for one day ruled completely by harmony and love, a kind of foretaste of the Messianic future."
[R Yehonatan Chipman]

According to Rabbi Chipman, Kagavna is explaining a change in nature that occurs when Shabbos enters. Shabbos is a small taste of what is to come in Achreis HaYamim.

"...the difference between the Mitnaggedic and Hasidic approach is epitomized in the texts read at this point in the Shabbat service. Mitnaggedim read Bameh Madlikin, a chapter from Mishnah listing various kinds of wicks and oils permitted and proscribed for use in the Shabbat lamps, and related rules. It is technical, formal, rooted in the rule of law and symbolizing, if you will, holiness acquired through halakhic discipline. Hasidim, by contrast, read Kegavna—a poetic invocation of Shabbat as a time of unity, of harmony, of banishment of all negative forces, a world-view which for a moment transcends and nullifies all differences—all united in an intense longing for kedushah, for holiness.”
[R Soloveitchik quoted by R Chipman]

--> Bameh Madlikin is representative of the halachic components surrounding shabbos. The Aseh and Lo Taaseh that make up the actual Mitzvah of Shabbos.

--> Kagavna is representative of the spiritual side of shabbos. The state of mind and the awareness of Kedusha. Shabbos is so much more than rules and rituals, we must incorporate the feeling of shabbos and recognize that we may not even grasp the intrinsic spirituality.

על הניגוד שבין הקטעים 'במה מדליקין' ו'כגוונא': מסופר על ויכוח בין חסיד למתנגד בשאלה מה ראוי לומר בערב שבת לפני 'ברכו'. אמר המתנגד: בשעה שאני אומר 'במה מדליקין' אני לומד לפחות פרק משניות מהלכות שבת ומשתדל להבין את הנאמר בו, ואילו אתה אומר 'כגוונא' בלי להבין אף מילה. ענה החסיד ואמר: אודה על האמת, אכן איני מבין את פשר המילים, אבל בשעה שאני אומר 'כגוונא' אני חש משק כנפי מלאכים סביבי, ואילו אתה האומר 'במה מדליקין' ועוסק בזפת ובשעווה, בשמן קיק ובחלב מבושל, נזכר מן הסתם בסחורה שבחנותך ושרוי כולך באווירה של חולין.
[Chanan Porat]

According to this story, the point of Kagavna is not to necessarily understood what is being said but to accept that there are aspects of Avodas Hashem that are beyond our human capacity to understand.