First off, let me specify that we don't cut off our penises, and neither do Muslims, who also have the rite of circumcision. Now, as to WHY God ordered Abraham to remove the foreskin of Isaac's penis at age 8 days, and at the same time remove the foreskin of Ishmael's penis (Ishmael was 13 years old; OUCH!), I don't know. The Bible simply says that God ordered Abraham to do it. He also had to do it to himself (MAJOR OUCH!). The Bible only says that God commanded it as a mark to indicate that Jews were in the covenant relationship with God.
Now, Jews are the children of Isaac, and have continued to perform this act to infants at the age of 8 days. Arabs are the descendants of Ishmael and continue to do this ceremony on boys at the age of 13 years. Now, Ishmael was NOT the son of the Promise, Isaac was. Hence the Bible said that Ishmael would become a great nation, which he did, founding the 12 Tribes of Arabia. So he was not neglected by God, but he was not the son of the Promise in the way that Isaac was.
But back to circumcision as a commandment. The Bible doesn't explain, why that as opposed to something else. Now, it is possible that the Talmud does. I have not examined the subject. I don't have the Talmud in my library, because I can't afford it, quite frankly. The whole collection of 20 odd books costs several hundred dollars. I do have a compendium of Jewish Law that might explain it, or at least attempt to. I shall do some research in my compendium, and perhaps visit the library at the shul and see what I can find in the Talmud. Perhaps the Rabbi can explain it to me as well. I shall tell you my answer if and when I get one.