Why do we sing "chad gadya" at the pesach seder? What is it supposed to be about?
Also, as a side question, is it "d'zabin aba" or "dizvan aba"?
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Although he does not go into detail about that last item, he states that the details are evident to the wise. If this is indeed the meaning of the song, it is almost like a miniature version of Shir Hashirim, which is (according to some) another version of the exact same allegory, and is read at the same point in the seder.
There is a footnote in The Practical Talmud Dictionary (p. 98) stating that the correct vocalization is "diz'van".
This is my own interpretation, (based on Jewish sources) but I like it.
Regarding d'zabin aba" or "dizvan aba", the former means that he sold while the latter means that he bought.
As to what it means, while it is possible to invest any text with deep meaning, my assumption would be that it is indeed an adaptation of a German children's song. I've noted the shift in verbs from interesting Aramaic ones in the beginning (in the stanzas paralleling the non-Jewish song) to boring Hebrew ones in the frum-ish conclusion.
They symbolism in the Vilna Goan's interpretation of Chad Gadya makes us realizes the song is about all of Jewish history. But as to why we sing the song (this insight comes from Aish Rabbi Sholom Denbo) think of the difference between history and memory. History is a bunch of facts from a book you learned in school that you barely remember. Memories are the experiences you yourself went though and have become the lenses you use to currently view the world. Your memories shape almost everything you interpret and almost every action you take.
On Passover we are supposed to relive the story of the Exodus. Not as a history story, but as a memory that we went through. That shaped us before and currently shapes us today. And then expand that awareness to the context of all of Jewish history to recognize where you are, and where you are going. The scope of Jewish history is encapsulated in Chad Gadya.
If you want to read a little more in depth about Chad Gadya check out this article on my website.
One insight that I have about this is that it comes to answer the famous question of why Paroah and the Egyptians were punished when the Galus was foretold 400 years earlier. The point brout out in the Chad Gadya is that if you toggle good-bad from the bad cat eating the goat and the good dog being upset about that and punishing it, you land up with the angel of death being good and why did God kill him?
The answer in this rhyme is self evident. Just because the cat sinned that has nothing to do with the dog. Here too, although there was a decree of Galus the Egyptians had no business getting involved.
Rabbi Jonathan Sacks in his Haggadah says that the "two coins" that Abba (Hashem) use to "purchase" (redeem from Mitzrayim) the gad (Bnai Yisrael) were Moshe and Aharon. He suggests the analogy of each of the attackers were the various empires who conquered us by attacking the previous rulers.
Ashur (cat), Bavel (dog), Persia (stick), Greeks (fire), Romans (water), Islam (ox), Christianity (slaughter), Ottoman Empire (Malach Hamaves).
Others use other connections for the last three. Some even use Worl War II for the Angel of Death. Of course, hashem redeems us all.
He also suggests that it is based on Rav Yehuda Bar Ilai in Baba Basra 10a giving the list of ten strong things in the world ending with tzedakah tatzil mimaves (Mishlei 10:2)
According to most scholars the Chad Gadya is predicated on Midrash Rabba (Gen. 38:13) which conveys the basic idea of the mishna in Avot (2:6) "because you drowned [another] you will be drowned, and ultimately the one that drowns you will be drowned" (what goes around comes around), namely, an expression of accountability and God's master plan.