Making Sense of Maggid
(ה) וְעָנִ֨יתָ וְאָמַרְתָּ֜ לִפְנֵ֣י ׀ יְהֹוָ֣ה אֱלֹהֶ֗יךָ אֲרַמִּי֙ אֹבֵ֣ד אָבִ֔י וַיֵּ֣רֶד מִצְרַ֔יְמָה וַיָּ֥גׇר שָׁ֖ם בִּמְתֵ֣י מְעָ֑ט וַֽיְהִי־שָׁ֕ם לְג֥וֹי גָּד֖וֹל עָצ֥וּם וָרָֽב׃ (ו) וַיָּרֵ֧עוּ אֹתָ֛נוּ הַמִּצְרִ֖ים וַיְעַנּ֑וּנוּ וַיִּתְּנ֥וּ עָלֵ֖ינוּ עֲבֹדָ֥ה קָשָֽׁה׃ (ז) וַנִּצְעַ֕ק אֶל־יְהֹוָ֖ה אֱלֹהֵ֣י אֲבֹתֵ֑ינוּ וַיִּשְׁמַ֤ע יְהֹוָה֙ אֶת־קֹלֵ֔נוּ וַיַּ֧רְא אֶת־עׇנְיֵ֛נוּ וְאֶת־עֲמָלֵ֖נוּ וְאֶֽת־לַחֲצֵֽנוּ׃ (ח) וַיּוֹצִאֵ֤נוּ יְהֹוָה֙ מִמִּצְרַ֔יִם בְּיָ֤ד חֲזָקָה֙ וּבִזְרֹ֣עַ נְטוּיָ֔ה וּבְמֹרָ֖א גָּדֹ֑ל וּבְאֹת֖וֹת וּבְמֹפְתִֽים׃ (ט) וַיְבִאֵ֖נוּ אֶל־הַמָּק֣וֹם הַזֶּ֑ה וַיִּתֶּן־לָ֙נוּ֙ אֶת־הָאָ֣רֶץ הַזֹּ֔את אֶ֛רֶץ זָבַ֥ת חָלָ֖ב וּדְבָֽשׁ׃

(5) You shall then recite as follows before your God יהוה: “My father was a fugitive Aramean. He went down to Egypt with meager numbers and sojourned there; but there he became a great and very populous nation. (6) The Egyptians dealt harshly with us and oppressed us; they imposed heavy labor upon us. (7) We cried to יהוה, the God of our ancestors, and יהוה heard our plea and saw our plight, our misery, and our oppression. (8) יהוה freed us from Egypt by a mighty hand, by an outstretched arm and awesome power, and by signs and portents, (9) bringing us to this place and giving us this land, a land flowing with milk and honey.

What about Shmot 1-13?

https://www.sefaria.org/Exodus.1-13

Rabbi Joseph B. Soloveitchik - Noraot Harav Vol. 8

מִצְוַת קְרִיאָה עַל הַבִּכּוּרִים – שֶׁנִּצְטַוִּינוּ בַּהֲבִיאֵנוּ הַבִּכּוּרִים לַמִּקְדָּשׁ לִקְרוֹת עֲלֵיהֶם הַכְּתוּבִים אֵלּוּ הַנִּזְכָּרִים בְּפָרָשָׁה זוֹ, וְהֵן, מֵאֲרַמִּי אֹבֵד אָבִי עַד הִנֵּה הֵבֵאתִי אֶת רֵאשִׁית פְּרִי הָאֲדָמָה אֲשֶׁר נָתַתָּה לִּי יְיָ. וְעַל זֶה נֶאֱמַר (דברים כו ה) וְעָנִיתָ וְאָמַרְתָּ לִפְנֵי יְיָ אֱלֹהֶיךָ וְגוֹ'. וְזֹאת הַמִּצְוָה יִקְרְאוּ זִכְרוֹנָם לִבְרָכָה (סוטה לב א) מִקְרָא בִּכּוּרִים. וּמִצְוַת הֲבָאָתָן כְּתַבְתִּיו בְּסֵדֶר מִשְׁפָּטִים בְּמִצְוַת הֲבָאַת הַבִּכּוּרִים (מצוה עב), וְנִכְפְּלָה הַמִּצְוָה לַהֲבִיאָן כְּמוֹ כֵן בְּפָרָשָׁה זוֹ, וּכְבָר נוֹדַע כִּי הַרְבֵּה מִן הַמִּצְוֹת נִכְפָּלוֹת בַּתּוֹרָה, וְכֻלָּן לְעִנְיָן וּלְצֹרֶךְ גָּדוֹל.
The commandment of recital over the first-fruits: That we commanded when bringing the first-fruits to the Temple to recite these verses in this section over them; and they are from, “My father was a wandering Aramean” (Deuteronomy 26:5), until “behold I have brought the first of the fruit of the land that the Lord gave me” (Deuteronomy 26:10). And about this is it stated (Deuteronomy 26:5), “And you will answer and you will say in front of the Lord, your God, etc.” And they, may their memory be blessed, called this commandment, (Sotah 32a), “the recital of the first-fruits.” I have written about the commandment of bringing them in the Order of Mishpatim (Sefer HaChinukh 91); and the commandment to bring them is likewise repeated in this section. And we already know that many of the commandments are repeated in the Torah; and all of them are for a great matter or necessity.
מִשָּׁרְשֵׁי הַמִּצְוָה. לְפִי שֶׁהָאָדָם מְעוֹרֵר מַחְשְׁבוֹתָיו וּמְצַיֵּר בִּלְבָבוֹ הָאֱמֶת בְּכֹחַ דִּבְרֵי פִּיו, עַל כֵּן, בְּהֵיטִיב אֵלָיו הַשֵּׁם בָּרוּךְ הוּא וּבְבָרְכוֹ אוֹתוֹ וְאֶת אַדְמָתוֹ לַעֲשׂוֹת פֵּרוֹת וְזָכָה לַהֲבִיאָם לְבֵית אֱלֹהֵינוּ רָאוּי לוֹ לְעוֹרֵר לִבּוֹ בְּדִבְרֵי פִּיהוּ וְלַחְשֹׁב כִּי הַכֹּל הִגִּיעַ אֵלָיו מֵאֵת אֲדוֹן הָעוֹלָם, וִיסַפֵּר חֲסָדָיו יִתְבָּרַךְ עָלֵינוּ וְעַל כָּל עַם יִשְׂרָאֵל דֶּרֶךְ כְּלָל, וְעַל כֵּן, מַתְחִיל בְּעִנְיַן יַעֲקֹב אָבִינוּ שֶׁחִלְּצוֹ הָאֵל מִיַּד לָבָן, וְעִנְיַן עֲבוֹדַת הַמִּצְרִיִּים בָּנוּ, וְהִצִּילָנוּ הוּא בָּרוּךְ הוּא מִיָּדָם, וְאַחַר הַשֶּׁבַח מְבַקֵּשׁ מִלְּפָנָיו לְהַתְמִיד הַבְּרָכָה עָלָיו, וּמִתּוֹךְ הִתְעוֹרְרוּת נַפְשׁוֹ בְּשֶׁבַח הַשֵּׁם וּבְטוּבוֹ זוֹכֶה וּמִתְבָּרֶכֶת אַרְצוֹ, וְעַל כֵּן צִוָּנוּ בָּרוּךְ הוּא עַל זֶה, כִּי חָפֵץ חֶסֶד הוּא.
It is from the root of the commandment [that it is] since a man arouses his thoughts and draws the truth in his heart with the power of the words of his mouth. Therefore, in that God did good to him, and in that He blessed him and his land to bear fruits, and he merited to bring the fruits to the House of our God; it is appropriate for him to arouse his heart with the words of his mouth and ponder that everything arrived to him from the Master of the universe, and he recount His kindnesses, may He be blessed, upon us and upon the people of Israel, more generally. Therefore, he begins with the subject of Yaakov, our father, whom God rescued from the hand of Lavan, and the subject of the slavery of the Egyptians over us, and His, blessed be He, rescuing us from their hand. And following the praise, he requests from Him to eternally bestow the blessing on him. And from the arousal of his soul with the praise of God and His goodness, he will merit that his land be blessed. Therefore, God commanded us about this, since He desires kindness.

Is there a common denominatore between the verses of Bikkurim and retelling the story of the Exodus?

לְפִיכָךְ אֲנַחְנוּ חַיָּבִים לְהוֹדוֹת, לְהַלֵּל, לְשַׁבֵּחַ, לְפָאֵר, לְרוֹמֵם, לְהַדֵּר, לְבָרֵךְ, לְעַלֵּה וּלְקַלֵּס לְמִי שֶׁעָשָׂה לַאֲבוֹתֵינוּ וְלָנוּ אֶת־כָּל־הַנִסִּים הָאֵלּוּ: הוֹצִיאָנוּ מֵעַבְדוּת לְחֵרוּת מִיָּגוֹן לְשִׂמְחָה, וּמֵאֵבֶל לְיוֹם טוֹב, וּמֵאֲפֵלָה לְאוֹר גָּדוֹל, וּמִשִּׁעְבּוּד לִגְאֻלָּה. וְנֹאמַר לְפָנָיו שִׁירָה חֲדָשָׁה: הַלְלוּיָהּ.

Therefore we are obligated to thank, praise, laud, glorify, exalt, lavish, bless, raise high, and acclaim He who made all these miracles for our ancestors and for us: He brought us out from slavery to freedom, from sorrow to joy, from mourning to [celebration of] a festival, from darkness to great light, and from servitude to redemption. And let us say a new song before Him, Halleluyah!

(ז) וְלָקַחְתִּ֨י אֶתְכֶ֥ם לִי֙ לְעָ֔ם וְהָיִ֥יתִי לָכֶ֖ם לֵֽאלֹהִ֑ים וִֽידַעְתֶּ֗ם כִּ֣י אֲנִ֤י יְהֹוָה֙ אֱלֹ֣הֵיכֶ֔ם הַמּוֹצִ֣יא אֶתְכֶ֔ם מִתַּ֖חַת סִבְל֥וֹת מִצְרָֽיִם׃ (ח) וְהֵבֵאתִ֤י אֶתְכֶם֙ אֶל־הָאָ֔רֶץ אֲשֶׁ֤ר נָשָׂ֙אתִי֙ אֶת־יָדִ֔י לָתֵ֣ת אֹתָ֔הּ לְאַבְרָהָ֥ם לְיִצְחָ֖ק וּֽלְיַעֲקֹ֑ב וְנָתַתִּ֨י אֹתָ֥הּ לָכֶ֛ם מוֹרָשָׁ֖ה אֲנִ֥י יְהֹוָֽה׃

(7) And I will take you to be My people, and I will be your God. And you shall know that I, יהוה, am your God who freed you from the labors of the Egyptians. (8) I will bring you into the land which I swore*swore Lit. “raised My hand.” to give to Abraham, Isaac, and Jacob, and I will give it to you for a possession, I יהוה.”
צֵא וּלְמַד מַה בִּקֵּשׁ לָבָן הָאֲרַמִּי לַעֲשׂוֹת לְיַעֲקֹב אָבִינוּ: שֶׁפַּרְעֹה לֹא גָזַר אֶלָּא עַל הַזְּכָרִים, וְלָבָן בִּקֵּשׁ לַעֲקֹר אֶת־הַכֹּל. שֶׁנֶּאֱמַר: אֲרַמִּי אֹבֵד אָבִי, וַיֵּרֶד מִצְרַיְמָה וַיָּגָר שָׁם בִּמְתֵי מְעָט, וַיְהִי שָׁם לְגוֹי גָּדוֹל, עָצוּם וָרָב.
Go out and learn what Lavan the Aramean sought to do to Ya'akov, our father; since Pharaoh only decreed [the death sentence] on the males but Lavan sought to uproot the whole [people]. As it is stated (Deuteronomy 26:5), "An Aramean was destroying my father and he went down to Egypt, and he resided there with a small number and he became there a nation, great, powerful and numerous."
וַיְהִי שָׁם לְגוֹי. מְלַמֵד שֶׁהָיוּ יִשְׂרָאֵל מְצֻיָּנִים שָׁם. גָּדוֹל עָצוּם – כְּמָה שֶּׁנֶּאֱמַר: וּבְנֵי יִשְׂרָאֵל פָּרוּ וַיִּשְׁרְצוּ וַיִּרְבּוּ וַיַּעַצְמוּ בִּמְאֹד מְאֹד, וַתִּמָּלֵא הָאָרֶץ אֹתָם.
"And he became there a nation" - [this] teaches that Israel [became] distinguishable there. "Great, powerful" - as it is stated (Exodus 1:7), "And the Children of Israel multiplied and swarmed and grew numerous and strong, most exceedingly and the land became full of them."
וָרָב. כְּמָה שֶּׁנֶּאֱמַר: רְבָבָה כְּצֶמַח הַשָּׂדֶה נְתַתִּיךְ, וַתִּרְבִּי וַתִּגְדְּלִי וַתָּבֹאִי בַּעֲדִי עֲדָיִים, שָׁדַיִם נָכֹנוּ וּשְׂעָרֵךְ צִמֵּחַ, וְאַתְּ עֵרֹם וְעֶרְיָה. וָאֶעֱבֹר עָלַיִךְ וָאֶרְאֵךְ מִתְבּוֹסֶסֶת בְּדָמָיִךְ, וָאֹמַר לָךְ בְּדָמַיִךְ חֲיִי, וָאֹמַר לָךְ בְּדָמַיִךְ חֲיִי.
"And numerous" - as it is stated (Ezekiel 16:7), "I have given you to be numerous as the vegetation of the field, and you increased and grew and became highly ornamented, your breasts were set and your hair grew, but you were naked and barren." "And when I passed by thee, and saw thee weltering in thy blood, I said to thee, In thy blood live! yea, I said to thee, In thy blood live!" (Ezekiel 16:6).
וַיָּרֵעוּ אֹתָנוּ הַמִּצְרִים וַיְעַנּוּנוּ, וַיִתְּנוּ עָלֵינוּ עֲבֹדָה קָשָׁה. וַיָּרֵעוּ אֹתָנוּ הַמִּצְרִים – כְּמָה שֶּׁנֶּאֱמַר: הָבָה נִתְחַכְּמָה לוֹ פֶּן יִרְבֶּה, וְהָיָה כִּי תִקְרֶאנָה מִלְחָמָה וְנוֹסַף גַּם הוּא עַל שֹׂנְאֵינוּ וְנִלְחַם־בָּנוּ, וְעָלָה מִן־הָאָרֶץ.
"And the Egyptians did bad to us" (Deuteronomy 26:6) - as it is stated (Exodus 1:10), "Let us be wise towards him, lest he multiply and it will be that when war is called, he too will join with our enemies and fight against us and go up from the land."