What does, 'A wandering Aramean was my father' mean in Dt 26:5?
Why was this response required to be made before the LORD?
Like many things, there are multiple possible answers to this question.
The pshat is that this is the first line of a speech given by a person bringing their bikurim (first fruits) as an offering to the temple. Take a look at Deut 26.5-10 and note the whole speech. It's basically a brief synopsis of Jewish history - our forefathers went down to Egypt, were enslaved, hashem brought us out and took us to Israel, therefore I am bringing this offering to hashem. (According to this interpretation, the Aramean was either Abraham or Yaakov - see below). The declaration is made in front of hashem and the cohen because this is the individual making an offering to hashem and making a public declaration of his attachment to hashem along with it.
The second (very well known) answer is the midrashic one that originates in the Mechilta. The Mechilta interprets this entire section midrashically (as I suppose one would expect from a midrashic source) and interprets this line as meaning "An Aramean tried to destroy my father", referring to Laban. The instructions for the Pesach seder in the Mishnah (Pesachim 10:4) says to drash from the line "Arami Oved Avi" until the end of the section. Ultimately the drash from the Mechilta became the commonly used text for this exercise, and since it's printed in the haggadah it has become extremely well known.
In case you're curious what some of the meforshim say, Rashbam says the Aramean was Abraham, and Seforno says the Aramean was Yaakov. Rashi quotes the midrashic interpretation and says the Aramean was Laban.
The standard Jewish interpretation of the verse that most everyone will know, as that is what Rashi says, and that is what the Passover Haggadah does with the verse, is that the Aramean is Lavan, who sought to destroy my father (Jacob).
A detour first about the root word here. אבד - means lost or destroyed. "Wandering" is fairly interpretive.
The Ibn Ezra relates that root to the idea of destitute (thus related to the destroyed) interpretats closer to the translation you are using. The point is that Jacob was destitute and poor in Aram (with Lavan) and then went to Egypt where he was further a stranger in small number - the point being to emphasize humble beginnings.
The Radak suggests relating to the idea of the great suffering that Yakov experienced while living with Lavan.
The point being that this emphasizes the great gift bestowed by G-d in giving the land for which the first fruits are being brought, and that this is not some family inheritance to which the bringer is rightfully entitled (see here).
Going back to the source for the Bikurim ritual, (Devarim 26.1-11) verse 10 tells us to end this Bikurim (First-fruits) Recitation with our definitive connection to the Land of Eretz Yisrael! The Torah is emphasizing that we are no longer landless subject to the whims and wages of a foreign master [Laban]: "you have changed my wages these ten times," B'rashith 31.41) but we, as Ya'acov's children are the object of HaShem's blessing in the Land of Eretz Yisrael, that which HaShem has sworn to our fathers and has given to us: Devarim's emphasis in 26.2, and 10 is on the land (ha'Adamah הָאֲדָמָה that HaShem chose, [Moriah] the place of sacrifice B'rashith 8.8, 13, 21; 22.14) which the Children of Ya'acov possess for an inheritance!
Devarim 26.2 "that thou shalt take of the first of all the fruit of the ground, which thou shalt bring in from thy land that the LORD thy God giveth thee;" here, the emphasis is on the land and the first-fruits which belongs to the Priest in the place HaShem chose. "pri haAdamah" - "which thou shalt bring in from thy land...."
"And it will be when you are come into the land... and possess it, etc."; "I profess this day unto the L-RD thy G-D, that I am come unto the land which the L-RD swore unto our fathers to give us.'"
Devarim 26.10 "And now, behold, I have brought the first of the fruit of the land, which Thou, O L-RD, hast given me.' And thou shalt set it down before the L-RD thy G-D, and worship before the L-RD thy G-D."
The Midrash changed "oved" to "ibed":
The midrash tells us that if you change the vowels from "oved" to "ibed," the meaning is changed from "wandering" to "destroyed." Thus, " A wandering Aramean was my father" will read, "An Aramean tried to destroy my father."
Ya'acov lived in Aram, while courting Rachel and working for her father Laban, an Aramean [viz, which we would call a modern day Syrian]. Consequently, in the Pesach Hagaddah, Laban is usually seen as the Aramean who would have sought to destroy Ya'acob, reminding us of the treachery of Laban, who tricked Ya'acov into marrying Leah before Rachel, then tricked him into twenty years of servitude, and finally tried to deny him his dowry. Therefore, Laban may be viewed as the symbol of everyone who has tried to destroy the Jewish People.
While Rashi accepted this reading, Ibn Ezra strongly rejected it, in favor of the interpretation that the verse refers to Ya'acov, who, when he was in Aram, was lost.
In reading the Midrash (Hagaddah) together with B'rashith 31.43, if one takes into account the accusation of Laban that Ya'acov stole Laban's idols, we can see how this accusation might suffice to "uproot everything" (as the penalty for theft of idols would be a death sentence); yet, we are left with the phrase "he went down into Egypt," so identifying who went down to Egypt tells us who the Aramean is.
"A wandering Aramean was my father [’arami ’oved avi]; he went down into Egypt and lived there as an alien, few in number, and there he became a great nation, mighty and populous." The Torah says, "he went down into Egypt..., and there he became a great nation...." This passuk serves to identify who the wandering Aramean is: Clearly, our Torah teaches that Laban returned to Aram (32.1) and Ya'acov went down to Egypt and [there] became a great nation (46.27).
Ibn Ezra, rejects the interpretation of the Midrash and says, Ya'acov was lost while in Aram! (Which I interpret to mean he was without roots in the land [of Paddan Aram], that is, he was a landless dweller in tents (he kept to his "family trade" and to tradition.
While it might make sense to reconcile the Hagaddah's Midrashic interpretation ("An Aramean attempted to destroy... etc.") with the Torah's statement, " A wandering Aramean was my father;" one cannot escape the Torah's context, "he went down into Egypt."
The Torah never tells us that Laban went down to Egypt, but only that Ya'acov became a Great Nation there (in Mitzrayim) AFTER wandering landless as a shepard in Paddan Aram....
In answer to the second question, "Why was this response required to be made before the L-RD?": Again, the Torah is emphasizing our definitive connection to the Land of Eretz Yisrael! HaAdamah, "the ground" is prefixed with the definitive article Hey (the) to focus our attention on the place HaShem chose!
I disagree with this translation. Let's look at the Hebrew text:
וְעָנִיתָ וְאָמַרְתָּ לִפְנֵי יְהוָה אֱלֹהֶיךָ, אֲרַמִּי אֹבֵד אָבִי, וַיֵּרֶד מִצְרַיְמָה, וַיָּגָר שָׁם בִּמְתֵי מְעָט; וַיְהִי-שָׁם, לְגוֹי גָּדוֹל עָצוּם וָרָב.
This verse is expounded upon in the Pesach Haggaddah. I don't have the exact text before me, but the gist of it is that "an Aramean tried to destroy my father" - this is Lavan the Aramean, who was worse than Pharaoh.
As for why this is said at this time, it seems to me to be a short history lesson, like "remember where you came from".