What Happens at Baptism?

Well, you go under the water. Besides that, let's see...

In baptism, there is an inward work that takes part in the heart of a believer. Public baptism, and all baptisms should be public in some way, are an acknowledgment of your faith in Christ. It is your testimony before the world that you are choosing to identify yourself with Him. In baptism, you are acknowledging that you have put off the old man and put on Christ. While it is an outward sign, there is also an inward change. The Holy Spirit performs a spiritual circumcision of the heart.

Circumcision in the Old Testament was a command (Gen 17:10). It was a foreshadowing of that which was to come.

Go check these passages:

Genesis 17:11
Genesis 17:13
Deuteronomy 30:5-6

And look at this one:
Circumcise therefore the foreskin of your heart, and be no more stiffnecked.
Deuteronomy 10:16

Circumcision of the foreskin was a sign of the covenant between God and man, so then is circumcision of the heart in baptism a sign of the New Covenant. The body of sin is 'cut away' with the old desires and lusts. the true sign of a believer in the New Covenant is a circumcised heart.

For he is not a [real] Jew who is only one outwardly and publicly, nor is [true] circumcision something external and physical. But he is a Jew who is one inwardly, and [true] circumcision is of the heart, a spiritual and not a literal [matter]. His praise is not from men but from God.
Romans 2:28-29 AMP

We must have a circumcised heart.

That circumcision without hands is performed at baptism.

In whom also ye are circumcised with the circumcision made without hands, in putting off the body of the sins of the flesh by the circumcision of Christ: Buried with him in baptism, wherein also ye are risen with him through the faith of the operation of God, who hath raised him from the dead. And you, being dead in your sins and the uncircumcision of your flesh, hath he quickened together with him, having forgiven you all trespasses;
Colossians 2:11-13