The pesukim in Parshas Emor which describe Rosh HaShana (23:24) reveal little about the nature of the holiday.
דַּבֵּר אֶל-בְּנֵי יִשְׂרָאֵל, לֵאמֹר: בַּחֹדֶשׁ הַשְּׁבִיעִי בְּאֶחָד לַחֹדֶשׁ, יִהְיֶה לָכֶם שַׁבָּתוֹן--זִכְרוֹן תְּרוּעָה, מִקְרָא-קֹדֶשׁ
We are not told what we are commemorating, why on this day, or how the ritual of shofar plays into the theme of the holiday. Ramban explains:
ולא פירש הכתוב טעם המצווה הזאת, למה התרועה, ולמה נצטרך זיכרון לפני השם ביום הזה יותר משאר הימים, ולמה יצווה להיותו מקרא קדש כלל. אבל מפני שהוא בחדשו של יום הכיפורים בראש החודש נראה שבו יהיה דין לפניו יתברך כי בם ידין עמים, בראש השנה ישב לכסא שופט צדק, ואחרי כן בעשרת הימים ישא לפשע עבדיו. נרמז בכתוב העניין כאשר נודע בישראל מפי הנביאים ואבות קדושים.
The Torah does not elaborate on Rosh HaShana because it relies on the explanation given a few pesukim later for our celebration of Yom Kippur as a day of atonement. Coming so close together, it stands to reason that Rosh HaShana is intimately connected with Yom Kippur as a day of judgment, and the blowing of shofar is part and parcel of how we merit repentance and atonement.
Given that we see from the pesukim themselves (and the theme is reinforced many places in Chazal) that shofar is integral to kapparah, Chasam Sofer (P' Nitzavim d"h v'chaim sha'alu) asks how Chazal could cancel shofar on Shabbos just because of the slight chance that someone would err and carry a shofar in the street. Does the slight risk of chilul Shabbos outweigh the great risk of not having complete atonement?
The answer must be that yes, for even a slight chance of chilul Shabbos it is worth postponing tekiyas shofar, and we have Chazal's assurance that we will suffer no harm and diminishment of kaparah for placing paramount our concern for Shabbos. Quite the contrary -- our demonstration of concern for Shabbos and respect for Chazal's gezeiros will increase our merits.
However, writes the Chasam Sofer, this is a double-edged sword. If a person on every Shabbos of the year treats Shabbos with less than the respect it deserves, it is very hard to justify not blowing shofar because of an ostensible concern for kedushas Shabbos. How can you suddenly be such a machmir on Shabbos when it comes to shofar when for your own needs you were out playing ball on Shabbos, when you dress in casual clothes that are less than fitting for kedushas Shabbos, when your speech on Shabbos revolves around business instead of Torah, etc. whatever your own example may be.
Rather than bemoan the mistreatment of Shabbos in the past as an obstacle, reflecting on the concern Chazal demonstrated for Shabbos can inspire us to commit ourselves to greater respect for the sanctity of Shabbos in the future.
See here for other thoughts on the issue of not blowing shofar on Shabbos (Meshech Chochma, Minchas Elazar). I have spent a little time trying to organize some of the past posts by topic, so for more of what we discussed in past years regarding Rosh HaShana just click the topic on the left side.