This is something that I learned in high school from my esteemed principal. I really liked the idea and so actually wrote it down for posterity . . . enjoy.
In the haggada, we use four pesukim to explain the story of the Exodus from Egypt - each pasuk is quoted and then every word in the pasuk is subsequently expanded upon:
אֲרַמִּי אֹבֵד אָבִי, וַיֵּרֶד מִצְרַיְמָה, וַיָּגָר שָׁם בִּמְתֵי מְעָט; וַיְהִי-שָׁם, לְגוֹי גָּדוֹל עָצוּם וָרָב.
An Aramean nearly caused my father to perish, and he went down into Egypt, and sojourned there, few in number; and he became there a nation, great, mighty, and populous.
וַיָּרֵעוּ אֹתָנוּ הַמִּצְרִים, וַיְעַנּוּנוּ; וַיִּתְּנוּ עָלֵינוּ, עֲבֹדָה קָשָׁה
And the Egyptians dealt ill with us, and afflicted us, and laid upon us hard bondage.
וַנִּצְעַק, אֶל-יְהוָה אֱלֹהֵי אֲבֹתֵינוּ; וַיִּשְׁמַע יְהוָה אֶת-קֹלֵנוּ, וַיַּרְא אֶת-עָנְיֵנוּ וְאֶת-עֲמָלֵנוּ וְאֶת-לַחֲצֵנוּ
And we cried unto the L-RD, the G-d of our fathers, and the L-RD heard our voice, and saw our affliction, and our toil, and our oppression.
וַיּוֹצִאֵנוּ יְהוָה, מִמִּצְרַיִם, בְּיָד חֲזָקָה וּבִזְרֹעַ נְטוּיָה, וּבְמֹרָא גָּדֹל--וּבְאֹתוֹת, וּבְמֹפְתִים
And the L-RD brought us forth out of Egypt with a mighty hand, and with an outstretched arm, and with great terribleness, and with signs, and with wonders.
Where, exactly, did these pesukim come from?
These are the phrases said upon fulfilling the mitzvah of bikkurim - of bringing the first of the newly blossomed fruits to the kohen gadol in the Beit Hamikdash, as explained in Parshat Ki Tavo, chapter 26, verses 5-8.
Whaaaaa? Why, when bringing a gift of fruits, do we go back and say "My forefathers were slaves in Egypt"? Seems to kill the moment, which is ostensibly one of happiness - the crops flourished, there will be food for everyone to eat - why are we dredging up historical memories of slavery and bondage?
Because contrary to what it may seem like at first glance, these are words that actually bring us joy. The mitzvah of bikkurim is an acknowledgement of Hashem, in His boundless munificence and glory, choosing us as a nation. And so we say precisely these pesukim as a way to thank Him! These pesukim recognize that Hashem redeemed us from slavery and elevated our status to that of His chosen nation. Thus, these four pesukim (and you'll notice how the four pesukim follows the pattern of "four" found in the haggada) are associated with the concept of hakarat hatov, of gratitude, to Hakadosh Baruch Hu for choosing the descendants of Yaakov to be His nation, the one that will accept the yoke and beauty of the Torah.
Using these pesukim to tell the story of Mitzrayim evokes that feeling of overwhelming gratitude to Hashem that we feel at the time of the bringing of the bikkurim. He redeemed us, He chose us - He infused us with the holiness of being the Jewish nation. By saying these lines, we bring the mood and essence of the bikkurim to our own tables: thanking G-d for what He has done for us.
Furthermore, a person who brings bikkurim is not someone who actually experienced the Yetziah, but rather, is a descendant of someone who did. However, these pesukim are in first person, as though the sayer of them is speaking about a personal experience. So too, we, living 3,320 years after the Exodus, relate the story of Egypt as though it happened to us. These pesukim connect us way back to the people who left Mitzrayim after the shibud, just as the person who is bringing bikkurim connects the events of the enslavement to the ultimate freedom experienced in Eretz Yisrael.
R' Yosef Dov Soloveitchik says the statement made when bringing bikkurim (ie the four pesukim quoted above) celebrates the status of being free. A slave is considered to have no yichus, no lineage - he is just a cog in the wheel. However, with these words, we connect to our family - "Arami oved avi" - my father Yaakov was nearly killed by his father-in-law Lavan. We establish our larger connection to all of the Jewish people with these words, and acknowledging the fact that we have lineage, ancestry, is part of the ultimate statement of freedom.
Have a chag kasher vesameach, and may this be our last Pesach experienced in galut!