The Rambam bases the mitzvah of Brit Milah –circumcision on the following verse in Tazria:

ויקרא יב (ג) וּבַיּ֖וֹם הַשְּׁמִינִ֑י יִמּ֖וֹל בְּשַׂ֥ר עׇרְלָתֽוֹ׃

Vayikra 12

3 And on the eighth day, the flesh of his foreskin shall be circumcised.

משנה תורה – הקדמה – מצות עשה – רטו  – לָמוּל אֶת הַבֵּן, שֶׁנֶּאֱמָר "וּבַיּוֹם, הַשְּׁמִינִי, יִמּוֹל, בְּשַׂר עָרְלָתוֹ" (ויקרא יב,ג)

Rambam - Mishna Torah Introduction - Positive Mitzvah 215

To circumcise a son, as [Leviticus 12:3] states: "On the eighth day, the flesh of his foreskin shall be circumcised.”

The Sefer HaChinuch discusses the ideas associated with this mitzvah:

ספר החינוך ב

משרשי מצוה זו, לפי שרצה השם יתברך לקבע בעם אשר הבדיל להיות נקרא על שמו, אות קבוע בגופם, להבדילם משאר העמים בצורת גופם, כמו שהם מבדלים מהם בצורת נפשותם, אשר מוצאם ומובאם איננו שוה. ונקבע ההבדל בגלת הזהב, לפי שהוא סבה לקיום המין, מלבד שיש בו תשלום צורת הגוף, כמו שאמרנו. והעם הנבחר חפץ השם יתברך להשלים תכונתו, ורצה להיות ההשלמה על ידי האדם ולא בראו שלם מבטן, לרמז אליו כי כאשר תשלום צורת גופו על ידו, כן בידו להשלים צורת נפשו בהכשר פעלותיו

Sefer HaChinuch 2

It is from the roots of this commandment [that it is] because God wanted to establish in His nation, that He separated to be called by His name, a permanent sign on their body; to separate them from the other nations in the form of their bodies just like they are separated from them in the form of their souls, the going out and coming in of which are not similar. He established this difference in the 'golden fountain,' because this is the reason for the existence of people, besides being a completion of the physical body as we mentioned. God wanted to complete His plan with the chosen nation. He wanted men to complete the creation of his body, as He did not create him complete from the womb; [so as] to hint to him that just like the completion of the form of his body is through him, so [too] is it in his hand to complete the form of his soul, by refining his actions.

The Sefer HaChinuch suggests that the foundation of the mitzvah of Brit Milah differentiates us as a distinct nation. He further explains that the mitzvah serves to complete the body, to make a final correction. It conveys to us that man does not come into this world perfected, and that he must pursue his own development and refine his own character through his actions. The underlying message iּs that man has work to do as soon as he arrives.

Rambam - Guide for the Perplexed III 49

As regards circumcision, I think that one of its objects is to limit sexual intercourse, and to weaken the organ of generation as far as possible, and thus cause man to be moderate. Some people believe that circumcision is to remove a defect in man's formation; but every one can easily reply: How can products of nature be deficient so as to require external completion, especially as the use of the fore-skin to that organ is evident. This commandment has not been enjoined as a complement to a deficient physical creation, but as a means for perfecting man's moral shortcomings. The bodily injury caused to that organ is exactly that which is desired; it does not interrupt any vital function, nor does it destroy the power of generation. Circumcision simply counteracts excessive lust; for there is no doubt that circumcision weakens the power of sexual excitement, and sometimes lessens the natural enjoyment…This is, as I believe, the best reason for the commandment concerning circumcision. And who was the first to perform this commandment? Abraham, our father! of whom it is well known how he feared sin; it is described by our Sages in reference to the words, "Behold, now I know that thou art a fair woman to look upon" (Gen. 12:11).

There is, however, another important object in this commandment. It gives to all members of the same faith, i.e., to all believers in the Unity of God, a common bodily sign…It is also a fact that there is much mutual love and assistance among people that are united by the same sign when they consider it as [the symbol of] a covenant. Circumcision is likewise the [symbol of the] covenant which Abraham made in connection with the belief in God's Unity. So also every one that is circumcised enters the covenant of Abraham to believe in the unity of God, in accordance with the words of the Law, "To be a God unto thee, and to thy seed after thee" (Gen. 17:7). This purpose of the circumcision is as important as the first, and perhaps more important. (Friedlander translation, p.267-268).

The Rambam’s approach differs from the Sefer HaChinuch in that circumcision does not correct or complete the body. In fact, it creates a limitation. But this limitation placed on the physical is indeed inline with the mitzvah, “as a means for perfecting man’s moral shortcomings”. We do not pursue boundless physical enjoyment. While the Torah’s philosophy approves of physical pleasure, it is easy for man to get lost in its pursuit and lose sight of his higher and more noble goals. This fundamental mitzvah is a covenant and statement of our values beyond the physical. Man must ultimately limit his physical pursuits and direct his energies toward his moral and spiritual development.

The Rambam also highlights the communal aspect of Brit Milah. It is a sign that ties our community together. It is interesting that this is also codified in the blessings and responses during the Brit Milah service:

ערוך השולחן יורה דעה רסה:ח

ולפי זה אני אומר דברכת "להכניסו" היא ברכה אחרת לגמרי, וזהו כאילו התינוק היה מברכה. והיינו שמקודם מברך "על המילה" כמו על כל המצות, ואחר כך מברכין על מה שצוה הקדוש ברוך הוא להכניס ילד ישראל מיום שנולד תחת כנפי השכינה, ולא ככל המצות שלא נצטוה האדם בה עד שנעשה גדול. ולא על עצם המילה מברכין ברכה זו, אלא שעל ידי מצוה זו נכנס נפש זה תחת כנפי השכינה, ונכנס בברית כלל ישראל, שזהו בריתו של אברהם אבינו שהקדוש ברוך הוא כרת עמו ברית זה. ולכן משיבין העולם: "כשם שנכנס לברית כן יכנס לתורה וכו' ", כלומר: שעיקר ההכנסה תחת כנפי השכינה הוא תורה ומעשים טובים, לכן מברכין אותו שיגיע לכלל זה. ואגב אומרים "לחופה" גם כן, וגם זהו מצוה.

Aruch HaShulchan - Yoreh Deah - 265:8

It is not the actual circumcision itself that we make this (second blessing), rather through this mitzvah this person enters under the wings of the Divine presence, and enters into the covenant with the Jewish community. This is the covenant of Avraham our forefather which the Holy One Blessed Be He established with him. This is why the group responds: “Just like he is entered into this covenant, so he will enter into Torah, marriage, and good actions”, meaning the essential joining under the wings of the Divine presence is through Torah and good actions.

It is notable that Halakha mandates a specific rejoinder by the community in response to the blessing on circumcision. The community response goes beyond a simple Amen, stating our ultimate values of Torah and beneficial actions that refine the human being and the world. The prominent community response is consistent with the understanding of the Rambam and the Sefer HaChinuch, that Milah is a tie between the individual and covenantal community.

References

Maimonides, M., & Friedlander, M. (1953). The guide of the perplexed of Maimonides. New York: Hebrew Publishing Co.

Maimonides, M., Touger, E., Kaplan, B., Abramson, Z., & Silverstein, S. (2007). Mishneh Torah. New York: Moznaim Pub. Retrieved from: https://www.chabad.org/library/article_cdo/aid/682956/jewish/Mishneh-Torah.htm

Rosenberg, A. (1980). A new English translation of the Hebrew Bible text and Rashi, with a commentary digest. New York: Judaica Press. Retrieved from: https://www.chabad.org/library/bible_cdo/

Sefer HaChinukh, translated by Nataf, Francis (2018). Sefaria. Retrieved from: https://www.sefaria.org/Sefer_HaChinukh.2.2

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