What did Jesus mean by “you must be born again” (John 3:7)?

Do not marvel that I said to you, ‘You must be born again.’ (John 3:7)

Jesus said this when talking to the Pharisee, Nicodemus. A few verses earlier we read

Jesus answered him, “Truly, truly, I say to you, unless one is born again he cannot see the kingdom of God.” Nicodemus said to him, “How can a man be born when he is old? Can he enter a second time into his mother’s womb and be born?” Jesus answered, “Truly, truly, I say to you, unless one is born of water and the Spirit, he cannot enter the kingdom of God. (John 3:3-5)

Jesus is referring to baptism where a person is immersed in water and rises a “new person”, just like being born again. Everyone who wants to be part of God’s kingdom must be baptised. This is repeated several times in the New Testament. For example,

Now when they heard this they were cut to the heart, and said to Peter and the rest of the apostles, “Brothers, what shall we do?” And Peter said to them, “Repent and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins, and you will receive the gift of the Holy Spirit. (Acts 2:37-38)