## kept growing ##

"grew up (and became a man)." The translation of this should make it clear that he was no longer a child when he lived out in the deserted regions.

## became strong in spirit ##

"became spiritually mature" or "developed a strong moral character" or "grew stronger in his relationship with God"

## until ##

This does not necessarily mark a stopping point. John  continued to live out in the desert even after he started preaching publicly. So it may be more clear to say "even up to the time when."

## appear publicly ##

This could be translated as "go before" or "preach in public."