The phrase comes from the Bible when Jesus answered a Pharisee called Nicodemus (John 3:1)
John 3:3 “Jesus answered and said to him, ‘Most assuredly, I say to you, unless one is born again, he cannot see the kingdom of God.” (NKJV)
Nicodemus was a Separatist who kept himself from liquor, immorality and anything he might do that would break the Old Testament Law. He was one of the most honourable and religious people of his day. In fact, John 3:10, Jesus refers to him as ‘The teacher’ in Israel (“Jesus answered and said to him, “Are you the teacher of Israel, and do not know these things?”).
Nicodemus came to Jesus with a burning question which he never got round to asking! It was obviously about getting into the kingdom of God. In spite of all his religious credentials, Jesus still tells him that in order to get into the Kingdom of God, he must be born again.
Salvation is not about ‘Religion’ or about ‘Church going’ it is about receiving ‘Life’. The issue is ‘Life’. Religion helps by setting out the doctrines and Church going helps by explaining the doctrines, but salvation is all about receiving life through Jesus. When receiving Jesus as your substitute for the penalty of sin, then the spirit within you (ie God consciousness – as opposed to soul which is self consciousness) that was once dead to God becomes alive, hence Born Again.
Someone once said, Christians are born twice but die once, whilst non-Christians are born once but die twice!
See also http://www.gotquestions.org/born-again.html