The mark of the covenant indicates the deep connection between us and the Holy One, blessed be He. Even when we are distant or in a dark place, God is with us and He lights up the darkness for us.

The meaning of the covenant is the connection between our souls and God:

This is the essence of the covenant of circumcision, which is called a covenant, a new [permanent] alliance between two parties who were previously distant. For the body is alienated [from the spiritual], and through this organ, closeness and connection are achieved. In truth, this organ is an exception that teaches us about all the rest: That the spiritual realm is close, but covered in darkness. Therefore, when permission is granted to remove the outer covering from this organ, its inner spiritual aspect is awakened…

The Holy One, blessed be He, is with us always:

The merit of circumcision always accompanies a man…and we should learn from this that the Holy One, blessed be He, walks everywhere with a person who serves God. (Rabbi Yehuda Aryeh Leib Alter, Sefat Emet, Lekh Lekha 5644)

An infant is circumcised when he is eight days old, just like our patriarch Isaac was circumcised at eight days old, and unlike Ishmael son of Abraham, who was circumcised at the age of thirteen. The difference between these two ages represents two different notions of the covenant. Do we enter the covenant out of understanding and consent, or is it a covenant of devotion, which is not limited to one’s level of knowledge or understanding?

The connection is formed at a moment when the infant is too young to understand or agree to the act:

Isaac was circumcised at the age of eight days. An eight-day-old infant is not considered to be giving consent [to entering into a covenant with God], and nevertheless he is bound at that point to the Holy One, blessed be He, with an unbreakable bond, as the verse states: “An eternal covenant” (Genesis 17:7).

The connection between us and the Holy One, blessed be He, is not limited to particular levels of understanding or intellect.

When establishing generations of Jews there is no place for a natural approach. The very essence and existence of the Jewish people are supernatural. The entire existence of a Jew, from his conception and birth, is through miracles, without any regard for nature [like Isaac, who was born when his parents were elderly]. The Holy One, blessed be He, said to our patriarch Abraham: True Jewish satisfaction comes only from a child whose birth and beginnings are caused by God, and who, when he is eight days old, at the first opportunity [as before this it is dangerous to circumcise a child], is bound to the Holy One, blessed be He, in an eternal covenant, although he still does not understand and there is no possibility of asking him. The Judaism of such a boy will not be limited by his knowledge or intellect, but will be imbued with devotion. (Rabbi Menaĥem Mendel Schneerson, Likutei Siĥot 1, pp. 20–21)

Further reading: For more on the covenant of circumcision accompanying a man wherever he goes, see the story of King David, A Concise Guide to the Sages, p. 145.