The passage explains that the ram Abraham saw caught in the thicket was one of the ten things created at twilight, and its horns and shofar carry deep mystical significance related to Isaac's spiritual strength, the sound heard at Mount Sinai, and Israel's redemption through the shofar blown on the New Year, which represents God's kingship established at creation and at the giving of Torah.
ועל דרך המדרש אחר נאחז בסבך בקרניו, אחר שישראל נאחזין בעבירות כל ימות השנה ומסתבכין בצרות נוטלין בראש השנה השופר של איל, אחר שישראל נאחזין בעבירות ומסתבכין במלכיות, מבבל למדי וממדי ליון ומיון לאדום, סופן להגאל בקרנו של איל, שנאמר (זכריה ט) וה' בשופר יתקע. ודעת רז"ל כי שופר של מתן תורה שכתוב בו וקול שופר חזק מאד, שופר אילו של יצחק היה. ובראש השנה שהוא יום הדין נצטוינו לתקוע בשופר, ואע"פ ששנינו כל השופרות כשרין, מצוה מן המובחר בשל איל. וידוע כי מאמרם זה שופר אילו של יצחק היה כונתו על מדתו של יצחק, לא על גופו ועצמותיו ממש, שהרי האיל הזה כולו היה עולה כליל עם קרניו וטלפיו, אבל החכמים ירמזו לנו על מדת הגבורה ששמענו מפיו אנכי ולא יהיה לך, והוא פחד יצחק אשר אנו מבקשים בעת צרה ואומרים עננו פחד יצחק, הוא הקול הנשמע במתן תורה, ואליו אנו רומזים בקול השופר שאנו תוקעין בראש השנה שהוא היום שנברא בו העולם. והתבונן כי יום מתן תורה תחלת קבלת מלכותו, וכן יום שנברא בו העולם תחלת מלכותו, לפי שאין מלך בלא צבא.
A Midrashic (Bereshit Rabbah 56,9) approach: the words אחר נאחז בסבך בקרניו, “after Israel has become enmeshed by its sins all year round, and as a result find themselves in deep trouble, they will blow the horn of the ram, i.e. the שופר on New Year’s Day appealing to G’d to remember the binding of Yitzchak.” Another allegorical interpretation: After Israel has become “enmeshed” in sins and as a result has been exiled amongst the “thicket,” the kingdoms of Babylon, Medes, Greece and Rome, they will ultimately experience redemption. This will be heralded by the blowing of the ram’s horn as we mention in our daily עמידה prayer in the paragraph commencing with the words תקע בשופר גדול לחרותנו, “blow the great horn to announce our freedom.” This prayer is based on Zechariah 9,14 who assured us that G’d Himself will blow this Shofar. Our sages (in Pirke d'Rabbi Eliezer at the end of chapter 31) believe that we are dealing here with an allusion to the Shofar that G’d was going to blow during the revelation at Mount Sinai (19,16). It is written of that Shofar that its sound was very strong (Exodus 19,16) and that it was the Shofar of the ram found during the binding of Yitzchak. We are commanded to blow the Shofar on New Year’s Day. Although, halachically speaking (Rosh Hashanah 26,1), a number of horns from a number of animals are acceptable for this ceremony, the best way to fulfill the commandment is by means of a ram’s horn. When the sages described this Shofar which was blown on the occasion of the revelation as “the Shofar of Yitzchak,” they referred to the virtue of Yitzchak, to his outstanding characteristic. They did not mean to imply that the horn used on that occasion (some 360 years after the binding of Yitzchak) was physically the same horn as the one belonging to the ram which Avraham slaughtered. In fact, this would have been impossible as the rules of the burnt-offering include that the entire animal be burned (except its skin and hair). The horns and hooves are included in the parts which need to be burned. Our sages alluded to the attributes of גבורה, power, we heard from G’d’s mouth directly when He addressed the entire assembly of the Jewish people and told them the first two of the Ten Commandments, i.e. “I am the Lord, etc.,” and “you must not have any other gods, etc.” G’d revealed this attribute at that time and it is called by our sages פחד יצחק, (compare Genesis 31,53). This attribute of G’d to which Yitzchak more than any other of our patriarchs related, is what we call upon in times of distress when we say עננו פחד יצחק, “answer us ‘attribute’ which Yitzchak related to with utmost reverence.” This is the voice of the soul which was audible at the time when the Torah was given and we allude to this voice when we blow the Shofar on New Year’s Day, the anniversary of the creation (completion) of the universe. You should understand that the day on which the Torah was given was the first day G’d’s kingdom had been accepted as such by the Jewish people, seeing that there cannot be a king without an army. [The Jewish people in their capacity as G’d’s people, are considered G’d’s “army.”]