John the Baptist is a Priest

Zachariah goes up to the altar of incense to offer incense, the Angel of the Lord, Gabriel, appears, and tells him his wife will bear him a son, who will call John, he disbelieves and in struck dumb. Unlike the majority of the Temple's priest Zachariah is "righteous".

He remains dumb until after John's birth, Elizabeth names him and Zachariah writes his agreement, the peoples amazement is twofold. Remember he is a priest, and so therefore is the infant John. The first reason for amazement is Zachariah's speech returns, and immediately he pronounces the blessing he should have done in the Temple after burning incense, he does so by announcing the Benedictus:

Blessed be the Lord God of Israel; because he hath visited and wrought the redemption of His people:
And hath raised up an horn of salvation to us, in the house of David his servant:
As he spoke by the mouth of his holy prophets, who are from the beginning:
Salvation from our enemies, and from the hand of all that hate us:
To perform mercy to our fathers, and to remember his holy testament,
The oath, which he swore to Abraham our father, that he would grant to us,
That being delivered from the hand of our enemies, we may serve him without fear,
In holiness and justice before him, all our days.
And thou, child, shalt be called the prophet of the Highest: for thou shalt go before the face of the Lord to prepare his ways:
To give knowledge of salvation to his people, unto the remission of their sins:
Through the bowels of the mercy of our God, in which the Orient from on high hath visited us:
To enlighten them that sit in darkness, and in the shadow of death: to direct our feet into the way of peace.

This should be a second reason for the peoples amazement, these words of his prophecy but they aren't reason is simply that Zachariah calls him John, a name unknown in his family.

So what is so amazing? As a priest Zachariah was landless, numbered along with widows and orphans, at a time when not everyone believed in the Resurrection, one lived for as long as memory of you was kept alive, it was kept alive in the land and in one's sons and sons keeping your name alive. Priests were landless, their name was not kept alive by association with a well or a field or winepress, but by their sons inheriting their name, hence the shock that John was not going to keep anyone's memory alive. John's naming is a sign of something new, a break from the past.
The landless priest dwells in the desert, again a sign that here is a priest who takes his ministery seriously. Priests were expected to be celibate when ministering in the Temple John's loins are girt with his belt, some have suggested he had a relationship to the Essenes who practiced both baptism and perpetual celibacy. He lives on the sweetness of wild honey, it is "wild" not cultivated, it is supplied by divine providence.
This priest divides, he separates the just from the unjust, the clean from the unclean. This seems to be real role for Jewish priests; the priest inspects people for leprosy, judge whether an animal is fit for sacrifice. The priest was expected one who discerns between that which is pleasing to God and that which is not.
He makes two important priestly acts of discernment: first, identifying what is good, that Jesus is the acceptable sacrifice pleasing to God, "Behold the Lamb of God...". Secondly identifying what is bad, this brings about his death; he condemns the incestuous relationship between Herod and Herodias. Priests were expected to be prophetic but had been compromised since the time of the Maccabees.
The Temple priests seem happy to gossip but John denounces clearly and without comprise identifying the good and holy and denouncing the impure and unholy.