You found it! The ram's horn plays a role in the story of the binding of Isaac, which is read on the second day of Rosh Hashanah.
To learn more about the meaning of the shofar, watch Professor William Kolbrener's video.
(שלישי) וַיָּבֹ֗אוּ אֶל־הַמָּקוֹם֘ אֲשֶׁ֣ר אָֽמַר־ ל֣וֹ הָֽאֱלֹהִים֒ וַיִּ֨בֶן שָׁ֤ם אַבְרָהָם֙ אֶת־הַמִּזְבֵּ֔חַ וַיַּֽעֲרֹ֖ךְ אֶת־הָֽעֵצִ֑ים וַיַּֽעֲקֹד֙ אֶת־יִצְחָ֣ק בְּנ֔וֹ וַיָּשֶׂ֤ם אֹתוֹ֨ עַל־הַמִּזְבֵּ֔חַ מִמַּ֖עַל לָֽעֵצִֽים: וַיִּשְׁלַ֤ח אַבְרָהָם֙ אֶת־יָד֔וֹ וַיִּקַּ֖ח אֶת־הַמַּֽאֲכֶ֑לֶת לִשְׁחֹ֖ט אֶת־בְּנֽוֹ: וַיִּקְרָ֨א אֵלָ֝יו מַלְאַ֤ךְ יְהֹוָה֙ מִן־ הַשָּׁמַ֔יִם וַיֹּ֖אמֶר אַבְרָהָ֣ם| אַבְרָהָ֑ם וַיֹּ֖אמֶר הִנֵּֽנִי: וַיֹּ֗אמֶר אַל־תִּשְׁלַ֤ח יָֽדְךָ֙ אֶל־הַנַּ֔עַר וְאַל־תַּ֥עַשׂ ל֖וֹ מְא֑וּמָה כִּ֣י| עַתָּ֣ה יָדַ֗עְתִּי כִּי־יְרֵ֤א אֱלֹהִים֙ אַ֔תָּה וְלֹ֥א חָשַׂ֛כְתָּ אֶת־בִּנְךָ֥ אֶת־יְחִֽידְךָ֖ מִמֶּֽנִּי: וַיִּשָּׂ֨א אַבְרָהָ֝ם אֶת־עֵינָ֗יו וַיַּרְא֙ וְהִנֵּה־אַ֔יִל אַחַ֕ר נֶֽאֱחַ֥ז בַּסְּבַ֖ךְ בְּקַרְנָ֑יו וַיֵּ֤לֶךְ אַבְרָהָם֙ וַיִּקַּ֣ח אֶת־הָאַ֔יִל וַיַּֽעֲלֵ֥הוּ לְעֹלָ֖ה תַּ֥חַת בְּנֽוֹ: וַיִּקְרָ֧א אַבְרָהָ֛ם שֵׁם־הַמָּק֥וֹם הַה֖וּא יְהֹוָ֣ה| יִרְאֶ֑ה אֲשֶׁר֙ יֵֽאָמֵ֣ר הַיּ֔וֹם בְּהַ֥ר יְהֹוָ֖ה יֵֽרָאֶֽה:
They came to the place that God had designated to him, and Avraham built the altar there and arranged the wood, and bound his son Yitzchak and placed him on the altar on top of the wood. Avraham reached out his hand and took the knife to slaughter his son. An angel of Adonoy called to him from heaven and said, “Avraham! Avraham!” and he said, “Here I am.” He [God] said, “Do not harm the boy, nor do anything to him; for now I know that you are one who fears God and have not withheld your son, your only one, from Me.” Avraham raised his eyes and beheld a ram after it had been caught in the thicket by its horns; and Avraham went and took the ram, and brought it up as a burnt-offering instead of his son. Avraham called the name of that place, “Adonoy will see”; as it is said [to] this day “On Adonoy’s mountain, He will be seen.”
Unique to the Musaf prayer on Rosh Hashanah are the three added paragraphs - Malchuyot (Kingship), Zichronot (Remembrance), and Shofarot (Shofar Blasts). The following three clues each relate to one of the three sections. Answers will be found in the Rosh Hashanah Machzor Ashkenaz in the Musaf prayer of the first day of Rosh Hashanah.
Clue #3 Which prayer, that is said in every prayer service, is recited as the first part of the Malchuyot (Kingship) section of the amidah? What do you think is the connection between kingship and this prayer?