The expression means that besides body, which can neither be born anew (John 3:4), nor be renovated in its earthly course, but naturally withers as the time passes, we possess a more fundamental aspect of our being, which can be called "soul", or "character", or "self", or "person", and which, albeit being created and natural, can participate in the Uncreated and Supra-natural, in the very Spirit of God, in the very Grace of God and, thus, get invested, "infected", saturated, clad (whatever verb you may choose for expressing the inexpressible mystical happening) in what is Divine and Uncreated, becoming "newly born" in this way.
To give an analogy: as a piece of iron attached to a magnet is magnetized, becoming, as it were, itself a magnet by participation, so also we, having received the Spirit of God in our created selves, become divinized through conscious and intentional participation and co-working with the Creator Spirit, re-creating ourselves into "new creation", investing ourselves with authority of being "sons of God" (John 1:12), brothers and sisters of His Only Begotten and, thus, uncreated natural Son, becoming His co-heirs by adoption, receiving the boldness and authority through the Spirit of sonship, who is the Holy Spirit, to call God "Father" (Romans 8:15).
And what is this new birth? What is this becoming the "new creation" or deification of our human essence? It means that we shall no more have the proclivity towards sin that we have due to the consequences of our fallen condition, to the effect that we do God's will out of sense and knowledge of duty, but in the depth of heart we feel unhappy for that, because the desire of heart inclines towards a sweeter prospect of performing sin (Romans 7:15-20), like in an English proverb "a good man dreams about what a bad man does". However, after the second birth in the Holy Spirit those very innermost depths of our sinful natures will be cleansed and transformed, transfigured through the working of Grace, so that we shall already start loving and desiring divine will as sweet and inviting, just as we desired the sweetness and invitation of sin before, in our "old man" state (Romans 6:6), and even more so, for life of Grace not only counterbalances the death of sin, but thoroughly eclipses it and vanquishes it altogether (1 Cor. 15:55-57). And thus we shall, as the newly borns in Spirit, have life in superabundance an excess (John 10:10), the "excess" meaning the divine life, that does not pertain to humans qua humans, but humans as partakers of God, which was closed even to the highest prophets (Luke 10:24), but now accessible to all through Jesus Christ.
Eventually, this birth is accessible in any age, even for children and even for very old people, for Spirit blows where He wishes and of course He wishes to come to those who themselves wish to accept the Spirit, who is not blocked by such a trifle and naughtiness as is the flux of time, wrinkles on a face or a natural loss of the "flower of youth", which is idolized now by mass media and advertising culture that knows not Spirit and is thoroughly anti-human as a matter of fact.
"An aged man is but a paltry thing,
A tattered coat upon a stick, unless
Soul clap its hands and sing, and louder sing
For every tatter in its mortal dress" (W. B. Yeats "Sailing to Byzantium") -
indeed, the old age can be even a better opportunity to not get dispersed in non-eternal interests and more thoroughly commune with the Spirit of Lord, to start singing the divine songs, during life, and after life also, for all eternity, for we are to inherit the Eternal Kingdom (Luke 12:32).