Jesus explained that to be born again does not mean to physically be reborn. Rather, it means to be born of water and spirit.
3 Jesus answered and said unto him, Verily, verily, I say unto thee, Except a man be born again, he cannot see the kingdom of God.
4 Nicodemus saith unto him, How can a man be born when he is old? can he enter the second time into his mother’s womb, and be born?
5 Jesus answered, Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God.
6 That which is born of the flesh is flesh; and that which is born of the Spirit is spirit. (John 3)
To be born of water is to repent and be baptized. This is the part of the process that is performed by the individual. Jesus said, “And again I say unto you, ye must repent, and be baptized in my name, and become as a little child, or ye can in nowise inherit the kingdom of God.” (3 Nephi 11:38)
To be born of spirit is to be bathed in the Holy Ghost. This is the part of the process that is performed by God. Jesus said, “...after that ye are baptized with water, behold, I will baptize you with fire and with the Holy Ghost;” (3 Nephi 12:1)
When one is born again, they—through a combination of personal repentance and an outpouring of God’s power—enter into a state where they are redeemed from the fall of Adam, a state they continue in as long as they do not commit sin:
25 And the Lord said unto me: Marvel not that all mankind, yea, men and women, all nations, kindreds, tongues and people, must be born again; yea, born of God, changed from their carnal and fallen state, to a state of righteousness, being redeemed of God, becoming his sons and daughters;
26 And thus they become new creatures; and unless they do this, they can in nowise inherit the kingdom of God. (Mosiah 27)