Why is the foreskin removed? God gave it men, did God also tell them to have it removed?
Firstly, we circumcise because it the sign specifically commanded in the Torah, by God to the Abraham and his descendants for entering the eternal covenant with God (Bereshit, 17). In return, God made Abraham the father of a multitude of nations. Made him exceedingly fruitful, into nations, and kings. God gave his decedents the entire land of Canaan for an everlasting posession, and is their God.
But why specifically the prepuce?
Circumcision is a chok (a law with an unclear reasons), however in the guide of the perplexed Rambam gives his own rationale for cutting off the prepuce specifically:
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: the organ necessarily becomes weak when it loses blood and is deprived of its covering from the beginning. Our Sages (Beresh. Rabba, c. 80) say distinctly: It is hard for a woman, with whom an uncircumcised had sexual intercourse, to separate from him. 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. xii. 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, so that it is impossible for any one that is a stranger, to say that he belongs to them. For sometimes people say so for the purpose of obtaining some advantage, or in order to make some attack upon the Jews. No one, however, should circumcise himself or his son for any other reason but pure faith; for circumcision is not like an incision on the leg, or a burning in the arm, but a very difficult operation. 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 connexion 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. xvii. 7). This purpose of the circumcision is as important as the first, and perhaps more important.
While the Rambam's second reason makes the connection that the circumcision is a covenant for believers in God's unity, it doesn't really answer why the prepuce is specifically cut off as sign of God's unity - the tip of your nose, or an ear-lobe could have been the sign.
I'd offer my own answer (so take it with a grain of salt), building upon the Rambam's second reason and in conjunction with his reason for other chukim (laws with unclear reasons), for why the circumcision is a sign of God's unity. The Rambam explains the reasons for some other chukim by showing that that they were idolatrous. He references an ancient book called "The Nabataean Agriculture" to show how many of our chukim are restrictions of idolatrous practices of the pagan society Abraham was from.
It must now be clear to you, and no room can be left for any doubt, that the prohibition of wearing garments of wool and linen, of using the fruit of a tree in the first three years, and of mixing diverse species, are directed against idolatry, and that the prohibition against adopting heathen manners serves to remove anything which leads to idolatry, as has been shown by us.
I would link circumcision to this, since the Phallus was a common idol whose worship dates back to prehistoric times and is still today a sign of fertility in some cultures. By injuring our phallus, we are actively denying the worthiness of the penis to be worshiped.
While other nations worship the phalluses, Jews cut their own phalluses as a sign of God's unity. Only Hashem is God, not stone or even flesh which can be cut in two.
God commanded Avraham to circumcise his sons and future male descendants. The obligation was derived from that command to apply to all fathers of male children. The relevant sources are laid out by Ramba"m here.
There are many mitzvos completion of which necessitates altering something's physical state. Here are some more examples: [binding and waving] the 4 designated plants on Sukos, gifting choice fleece to priests, preparing olive oil for priestly annointment
The Midrash records a discussion of this issue (from vbm-torah.org). R' Akiva explains that God created the world and man so that man can improve it:
"Once the evil [Roman governor] Turnus Rufus asked Rabbi Akiva, 'Whose deeds are greater - God's or man's?' He replied, 'Man's deeds are greater.' Turnus Rufus asked him, 'Is man then capable of creating heaven and earth, or anything like them?' Rabbi Akiva replied, 'I was not referring to the sphere beyond man's ability, over which he has no control. I refer to those creations of which man is capable.' He then asked, 'Why do you circumcise yourselves?' Rabbi Akiva replied, 'I knew that that was the point of your question, and therefore I answered in the first place that man's deeds are greater than God's.' Rabbi Akiva brought him grains of wheat and some bread, and said: 'These grains of wheat are God's handiwork, and the bread is the handiwork of man. Is the latter not greater than the former?' Turnus Rufus answered him, 'If God wanted you to perform circumcision, why did He not create the child already circumcised while still in the womb?' Rabbi Akiva answered, 'Why do you not ask the same question concerning the umbilical cord, which remains attached to him and which his mother must cut? In response to your question - the reason why he does not emerge already circumcised is because God gave Israel the commandments in order that they would be purified by performing them. Therefore David wrote, 'Every word of God is pure (or, purified).'"
The idea mentioned in the moreh nevuchim can also be understood symbolically. While circumcision not actually limit pleasure, it symbolizes subjugating everything, including basic desires, to God's will. Judaism brought to the world Ethical Monotheism which called for higher ideals than just gaining power, such as helping the weak and the poor. Similarly, Judaism calls for circumcision as a symbol of rising above base desires.
I have understood the bris milah to be illuminated in Devarim 30:6
וּמָל יְהוָה אֱלֹהֶיךָ אֶת-לְבָבְךָ, וְאֶת-לְבַב זַרְעֶךָ: לְאַהֲבָה אֶת-יְהוָה אֱלֹהֶיךָ, בְּכָל-לְבָבְךָ וּבְכָל-נַפְשְׁךָ--לְמַעַן חַיֶּיךָ.
We might consider this factors deeply with teshuvah, in that the sin (the part that impedes reciprocity with the light of hashem) is "cut off" so the nation may live.