Milah rectifies the body and reveals the soul

פתח האהל פתח טוב פתחת כו' כי המילה הוא פתח הגוף כמ"ש בזוה"ק פ' בא דכתיב וישלחהו לעבוד האדמה אשר לוקח משם והוא הכתנות עור חלק הגוף הבא מן האדמה וע"י המילה נתקן הגוף כמ"ש במדרש תנחומא זרע יעבדנו הוא אחר המילה.

"The entrance of the tent" (Bereishis 18:1) — "a good entrance You have opened" — for the milah (circumcision) is the "entrance" of the body, as is written in the Zohar HaKadosh on parshas Bo: as it says, "and He sent him to work the ground from which he was taken" (Bereishis 3:23) — and this is "the garments of skin" (Bereishis 3:21), the bodily portion that comes from the earth; and through the milah the body is rectified, as is written in the Midrash Tanchuma, "seed shall serve Him" (Tehillim 22:31) — this is after the milah.

The Sefas Emes reads "the entrance of the tent" (where Avraham sat after his bris) as alluding to milah, which opens the "doorway" of the body. The body, made of earth — the "garments of skin" given after Adam's sin — is coarse and concealing. Circumcision rectifies that earthy body, so that one becomes fit to "serve Him," as the Midrash applies the verse to the state after milah.

ואז נפתח ונתגלה הארת הנשמה.

And then the illumination of the neshamah is opened and revealed.

Once the body is rectified through milah, the "entrance" opens and the light of the soul, previously concealed by the coarse body, shines forth.

ואיתא במשנה שלשה דברים מתלמידיו של אברהם אבינו ע"ה עין טובה ורוח נמוכה ונפש שפילה.

And it is taught in the Mishnah: three traits mark the disciples of Avraham Avinu — a good eye (ayin tovah), a humble spirit (ruach nemuchah), and a lowly soul (nefesh shefeilah) (Avos 5:19).

The Mishnah lists the three defining middos of Avraham's disciples: generosity of vision, humility of spirit, and a self-effacing soul.

ונראה שבזכות אלה המדות שבחר בהם אאע"ה ובדרך שאדם רוצה לילך מוליכין אותו ניתן לו ברית מילה חיתוך הוא נפש שפילה שמחסר מקצת נפש.

And it appears that in the merit of these middos that Avraham Avinu chose, and "in the way a person wishes to go, he is led" (Makkos 10b), he was given the bris milah: the cutting (chituch) is the "lowly soul," for it diminishes a part of the [bodily] self.

Because Avraham embraced these humble traits, and Heaven leads a person along the path he chooses, he merited the mitzvah of milah — which itself embodies those very middos. The cutting away corresponds to "a lowly soul," reducing a measure of physical self.

ורוח נמוכה פריעה.

And "a humble spirit" is the periah (uncovering).

The second stage of the bris, periah, corresponds to "a humble spirit."

ועין טובה היא ההארה שמתגלה אחר המילה שע"ז אמרו סו"ד ה' ליריאיו ע' אורות שמתגלה ע"י המילה סוד גי' שבעים כידוע:

And "a good eye" is the illumination that is revealed after the milah, of which they said, "The secret (sod) of Hashem is to those who fear Him" (Tehillim 25:14) — the seventy lights that are revealed through the milah, for "sod" has the gematria of seventy, as is known.

The third trait, "a good eye," corresponds to the spiritual light revealed once the bris is complete. "Sod" (secret) equals 70 in gematria, alluding to the seventy lights of Divine intimacy that milah unlocks — "the secret of Hashem is to those who fear Him." Thus the three stages and effects of the bris embody Avraham's three middos.

Summary: "The entrance of the tent" alludes to milah, the "doorway" of the body. The earthy, coarse body — the post-sin "garments of skin" — is rectified through circumcision, opening the way for the soul's light to shine. Because Avraham chose the three middos of his disciples (a good eye, a humble spirit, a lowly soul), and a person is led on the path he chooses, he merited milah, which embodies those very traits: the cutting is "a lowly soul," periah is "a humble spirit," and the revealed light afterward is "a good eye" — the seventy lights of "the secret of Hashem," since "sod" equals seventy.