Rav Chaim Kanievsky explains that the גדי is the Jewish people, as we are referred to in ספר ירמיהו שה פזורה, scattered sheep. The Father is ‘ה who took us as His people when He gave us the two לוחות.

וְאָתָא שׁוּנְרָא, וְאָכְלָה לְגַדְיָא, דְזַבִּין אַבָּא בִּתְרֵי זוּזֵי, חַד גַּדְיָא, חַד גַּדְיָא.

The גר”א explains this stanza as follows:
יעקב gave the main ברכה to יוסף who was then sold to מצרים because of the jealousy of his brothers which is represented by the cat because of its natural jealousy.

The ברכת השיר explains this stanza as follows: The שונרא, the cat, represents the wife of פוטיפר who tried to seduce Yosef. Yaakov  prophesied this without realizing when he said יוסף has been ripped apart and eaten by a wild animal,“the cat”.

The מעשה נסים explains this stanza as follows. That the שונרא was נבוכדנצר who was known in the מדרש as שונא רע (evil enemy), hence שונרא who destroyed the first בית המקדש.

Rav Chaim Kanievsky explains this stanza as follows:
The שונרא was נבוכדנצר and the Babelonians who the Navi Yeshayahu connects cats to Bavel as cats howl in their mansions.

וְאָתָא כַלְבָּא, וְנָשַׁךְ לְשׁוּנְרָא, דְּאָכְלָה לְגַדְיָא, דְזַבִּין אַבָּא בִּתְרֵי זוּזֵי, חַד גַּדְיָא, חַד גַּדְיָא.
The גר”א explains this stanza as follows:
יוסף and his brothers went down to מצרים and were then under the control of פרעה “the dog”.

The ברכת השיר explains this stanza as follows: The dog is פרעה who at this point in history recognized his master as
‘ה when he said to Yosef, ” ‘ה explained my dreams to you”.

The מעשה נסים explains this stanza as follows:
The dog represents מלך כורש because he nursed off of a dog. He killed בלשצר the king of בבל and eventually allowed the rebuilding of בית שני hence the dog bit the cat.

Rav Chaim Kanievsky explains this stanza as follows:
Nevuchadnetzar’s descendents were Kasdim not Babelonian who took over his kingdom. and Balshatzar, who was the grandson of Nevuchadnetzar, served his dogs from the vessels of the בית המקדש.

וְאָתָא חוּטְרָא, וְהִכָּה לְכַלְבָּא, דְּנָשַׁךְ לְשׁוּנְרָא, דְּאָכְלָה לְגַדְיָא, דְזַבִּין אַבָּא בִּתְרֵי זוּזֵי, חַד גַּדְיָא, חַד גַּדְיָא.

The גר”א explains this stanza as follows:
The stick represents the staff of משה and the miracles that were performed, that literally beat the dog, פרעה and the מצרים.

The ברכת השיר also expounds this stanza as follows:
The stick was the staff of משה that was used to punish the dog when he forgot who his master is.

The מעשה נסים explains this stanza as follows:
The stick represents the kingdom of יון who ruled by the staff and conquered פרס and later  ruled ארץ ישראל. This led to the story of חנוכה.

Rav Chaim Kanievsky explains this stanza as follows: The stick are the Medes, because Darius smashed Balshatzar’s head with a stick. It was the Medes who took the kingdom from the Kasdim.

וְאָתָא נוּרָא, וְשָׂרַף לְחוּטְרָא, דְהִכָּה לְכַלְבָּא, דְּנָשַׁךְ לְשׁוּנְרָא, דְּאָכְלָה לְגַדְיָא, דְזַבִּין אַבָּא בִּתְרֵי זוּזֵי, חַד גַּדְיָא, חַד גַּדְיָא.

The גר”א explains this stanza as follows: The staff of משה represented the revealed miracles of ‘ה, which lasted until the destruction of the first בית המקדש which was caused by the יצר הרע. The יצר הרע for idol worship is compared to fire which burnt up the staff and put an end to the obvious miracles.

The ברכת השיר explains this stanza as follows:
The staff showed the strength of Jewish people. The fire was the יצר הרע that destroyed the strength.

The מעשה נסים explains this stanza as follows:
The fire represents the נס of the נרות of the מנורה which marked the end of Greek control over ארץ ישראל.

Rav Chaim Kanievsky explains this stanza as follows:
The Persians took the power from the Medes. Achashverosh the king of Persia was described as having a a fiery coal in his stomach, hence the fire that burnt up the stick.

וְאָתָא מַיָּא, וְכָבָה לְנוּרָא, דְּשָׂרַף לְחוּטְרָא, דְהִכָּה לְכַלְבָּא, דְּנָשַׁךְ לְשׁוּנְרָא, דְּאָכְלָה לְגַדְיָא, דְזַבִּין אַבָּא בִּתְרֵי זוּזֵי, חַד גַּדְיָא, חַד גַּדְיָא.
The גר”א explains this stanza as follows, Water represents תורה and it was the תורה in the תפילה of the אנשי כנסת הגדולה that extinguished the fire of the יצר הרע for עבודה זרע.

The ברכת השיר explains this stanza as follows. The water resembles the תורה which conquers the יצר הרע as the גמרא states בראתי יצר הרע בראתי לו תורה תבלין meaning ה“ created the יצר הרע and he created the תורה as its cure.

The מעשה נסים explains this stanza as follows. The water represents Rome which states in the מדרש thaat Rome is called קול מים רבים which means the sound of mighty water. That echoed throughout the world and extinguished the fire of the חשמונים.

Rav Chaim Kanievsky explains this stanza as follows. Alexander the Macedonian conquered the Persians. He is referred to as water because of a Gemara in Maseches Tamid that says he washed his face in the waters of Gan Eden.

וְאָתָא תוֹרָא, וְשָׁתָא לְמַיָּא, דְּכָבָה לְנוּרָא, דְּשָׂרַף לְחוּטְרָא, דְהִכָּה לְכַלְבָּא, דְּנָשַׁךְ לְשׁוּנְרָא, דְּאָכְלָה לְגַדְיָא, דְזַבִּין אַבָּא בִּתְרֵי זוּזֵי, חַד גַּדְיָא, חַד גַּדְיָא.
The גר”א explains this stanza as follows:
The ox is אדום, or Rome, which eliminated the אנשי כנסת הגדולה.

The ברכת השיר explains this stanza as follows: The nation that took תורה away from Jewish people.

The מעשה נסים explains this stanza as follows: The ox is the kingdom of ישמעאל that ruled over ארץ ישראל.

Rav Chaim Kanievsky explains this stanza as follows: The ox is a reference to the Greeks who inherited the Macedonian Empire. They are called the ox because they forced the Jewish people to write on the horns of their oxen that we have no part of the G-D of Yisroel.

וְאָתָא הַשׁוֹחֵט, וְשָׁחַט לְתוֹרָא, דְּשָׁתָא לְמַיָּא, דְּכָבָה לְנוּרָא, דְּשָׂרַף לְחוּטְרָא,דְהִכָּה לְכַלְבָּא, דְּנָשַׁךְ לְשׁוּנְרָא, דְּאָכְלָה לְגַדְיָא, דְזַבִּין אַבָּא בִּתְרֵי זוּזֵי, חַד גַּדְיָא, חַד גַּדְיָא.

The גר”א explains this stanza as follows:
The שוחט will be משיח בן יוסף who will annihilate Rome.

The ברכת השיר explains this stanza as follows:
This is משיח בן יוסף who is a descendant of ירבעם and he destroys Rome.

The מעשה נסים explains this stanza as follows: Unlike everyone else, משיח בן יוסף does not conquer Rome but conquers ישמעאל because the last גלות of the Jewish people is called גלות ישמעאל.

Rav Chaim Kanievsky explains this stanza as follows:
This is the Romans that the Gemara says they slaughtered the mighty like ownerless calves.

וְאָתָא מַלְאַךְ הַמָּוֶת, וְשָׁחַט לְשׁוֹחֵט, דְּשָׁחַט לְתוֹרָא, דְּשָׁתָא לְמַיָּא, דְּכָבָה לְנוּרָא, דְּשָׂרַף לְחוּטְרָא, דְהִכָּה לְכַלְבָּא, דְּנָשַׁךְ לְשׁוּנְרָא, דְּאָכְלָה לְגַדְיָא, דְזַבִּין אַבָּא בִּתְרֵי זוּזֵי, חַד גַּדְיָא, חַד גַּדְיָא.

The גר”א explains this stanza as follows:
משיח בן יוסף dies in battle after conquering אדום.

The ברכת השיר explains this stanza as follows: משיח בן יוסף is allowed to be killed by the מַלְאַךְ הַמָּוֶת because of the sins of ירבעם his ancestor who separated the Jewish people. Therefore, the death of משיח בן יוסף will bring a final כפרה,  forgiveness, for the Jewish people

The מעשה נסים explains this stanza as follows: משיח בן יוסף does not die by the hands of man, but rather by the hand of מַלְאַךְ הַמָּוֶת himself.

Rav Chaim Kanievsky explains this stanza as follows:
The מַלְאַךְ הַמָּוֶת in this case is the descendents of Yishmael who rule after the descendents of Rome. They are referred to as the מַלְאַךְ הַמָּוֶת because they kill very easily.

וְאָתָא הַקָּדוֹשׁ בָּרוּךְ הוּא, וְשָׁחַט לְמַלְאָךְ הַמָּוֶת, דְּשָׁחַט לְשׁוֹחֵט, דְּשָׁחַט לְתוֹרָא, דְּשָׁתָא לְמַיָּא, דְּכָבָה לְנוּרָא, דְּשָׂרַף לְחוּטְרָא, דְּהִכָּה לְכַלְבָּא, דְּנָשַׁךְ לְשׁוּנְרָא, דְּאָכְלָה לְגַדְיָא, דְזַבִּין אַבָּא בִּתְרֵי זוּזֵי, חַד גַּדְיָא, חַד גַּדְיָא.

The גר”א explains this stanza as follows:
‘ה will bring everything back to the original ברכה that יצחק gave יעקב when יעקב brought him the two goats.

The ברכת השיר explains this stanza as follows: Eventually הַקָּדוֹשׁ בָּרוּךְ הוּא will destroy מַלְאַךְ הַמָּוֶת and blot out death forever. 

The מעשה נסים explains this stanza as follows: הַקָּדוֹשׁ בָּרוּךְ הוּא will redeem us from our troubles and all death will be swallowed up eternally.

Rav Chaim Kanievsky says the same thing:
The Kingdom of Hashem ruled by משיח will rise and the the descendents of Yishmael will fall. אמן ואמן

This is only a minor sampling of the many explanations of this awesome poem. Both in depth and in breadth. The one area in which they all concur is that in the end, all pain and suffering will cease with the final redemption.

May this come speedily in our days!

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