Where did John the Baptist get the authority to baptize?

According to D&C 84: 23-28 (http://scriptures.lds.org/en/dc/84), after
the Melchizdek Priesthood was taken from the earth with Moses, the
Aaronic Priesthood remained, even up to the time of John the Baptist.
His father held the priesthood and as the oldest son of Zacharias John
would also receive it. Joseph Smith said: "The Levitical Priesthood
is forever hereditary – fixed on the head of Aaron and his sons
forever, and was in active operation down to Zacharias the father of
John" (TPJS, p. 319).

John was filled with the Holy Ghost from his womb (v.27). When Mary
who was carrying the unborn Savior visited Elizabeth, John leapt in
her womb.
John was baptized as a child (v. 28) and ordained by an angel at eight
days old to become an Elias, the forerunner to the Savior.

Although Zacharias was killed by Herod's soldiers, before his death he
must have baptized his son and ordained his son to the Aaronic
priesthood. Zacharias and Elizabeth were likely witnesses to their
son's ordination by an Angel of God to the calling of Elias. It may
have been about the same time that he was circumcised, received his
name and a blessing from his father (Luke 1:59).

The work of John the Baptist

Isaiah 40:3-5 prophetically describes the mission of John the Baptist
who would "prepare the way of the Lord" before both his mortal
ministry and his second coming. Further John was to "make straight in
the desert a highway for our God [that] every valley shall be exalted,
and every mountain and hill shall be made low and the crooked shall be
made straight and the rough places plain." Roads in ancient Israel
and even today are sometimes crooked and they go up and down with the
terrain. When a high way is straightened and the dips are removed by
raising the valleys and lowering the mountains and hills, it becomes
much easier to travel upon. In the section of Missouri where I live
the roads twist and go up and down. It requires slowing down at every
turn. However, across the border in Iowa the roads are straight and
the dips have been removed.

Such was the mission of John in preparing for the first coming of
Jesus Christ. He prepared the road so that when the Savior began his
ministry in the Meridian of Time there was already a following and
many had been baptized. Similarly, in this age, on May 15, 1829 John
the Baptist came to the prophet Joseph Smith and Oliver Cowdery and
conferred upon them the Aaronic Priesthood and gave them the power to
baptize (see D&C 13). This began the great preparatory work for the
Lord's Second Coming, which is yet to occur. (See also 1 Nephi
10:7-10.)