Kad HaKemach: Idol worshippers would tattoo themselves to mark their devotion and commitment to their gods. We, the children of Jacob, have the Brit Milah as a sign that we belong to God and are descendants of Abraham, who sanctify our lives to God’s name.
Abarbanel (Genesis 17): The Brit Milah is a true covenant between God and Israel. God commits Himself, so to speak, to attach His Presence and provide Divine Providence to the descendents of Abraham, and we commit to always carry His Name with us, in our actions, speech and thoughts.
Tosefta, Nedarim: Rebbi taught: “How great is the Mitzvah of Milah! Abraham was not called ‘Tamim” – perfect – until he was circumcised.”
Tanchuma Yashan, Lech #25: Abraham was not told that he could “Walk before Me” until he had a Brit Milah. He was only able to walk with God, in a dependent manner, not before God, with independence and creativity. (Machberes Avodas Hashem)
Zohar (Volume 1) Fortunate is Israel that they can make an offering to God, equal and beyond the offerings in the Temple. The father and mother make an offering to God when then give their son to be circumcised, and they enter the Courtyard of the “Tuv” – The Best Abundance – of God, the place that is called Tzaddik – The Righteous One.
Eileh HaMitzvot of Maharam Hagiz: The Brit Milah weakens the power of the Evil Inclination that resides in the Gid Hanashe – The Sciatic Nerve – damaged by the angel of Esau, which limits a person’s ability to grow and, because of Jacob’s wound, can cause more damage than any other part of the Evil Inclination. (Machberes Avodas Hashem)
Rabbeinu Bachya (Genesis 17): There are three Mitzvot that are designated as “signs”: Milah, Shabbat and Tefillin. The three testify to God’s Unity and our faith in the Only God.
Maimonides, Guide For The Perplexed (3:49): We constantly see how important it is for people to have a sign or symbol that declares their unity. (Like a sports team’s uniform) Brit Milah is the sign all Jews carry, to share our unified commitment to God. The sign is hidden, as what we share, is internal, private and precious. (Machberes Avodas Hashem)