Who Josephus was
Josephus was a Jewish priest and military commander. He fought against Rome in the war of AD 66, was captured, and went over to the Roman side. In Rome, around AD 93, he wrote a huge history of the Jewish people for Roman readers: the Antiquities of the Jews.
He wasn't a Christian. He wasn't writing to support the Gospels. He was writing history.
What he says about John
In the middle of a passage about King Herod Antipas losing a war, Josephus stops to explain why many Jews thought Herod deserved it:
"Now some of the Jews thought that the destruction of Herod's army came from God... as a punishment of what he did against John, that was called the Baptist: for Herod slew him, who was a good man, and commanded the Jews to exercise virtue... and so to come to baptism."
Then he tells what happened to John:
"He was sent a prisoner, out of Herod's suspicious temper, to Macherus... and was there put to death."
Macherus was Herod's desert fortress east of the Dead Sea.
What the Gospels say
Now read the Gospels:
"John was in the desert baptizing, and preaching the baptism of penance for the remission of sins. And there went out to him all the country of Judea, and all they of Jerusalem, and were baptized by him in the river of Jordan."
Mark 1:4 to 5
"For Herod himself had sent and apprehended John, and bound him in prison... But sending an executioner, he commanded that his head should be brought in a dish. And he beheaded him in the prison."
Mark 6:17 and 27
Two angles, one man
The two accounts come at John from different directions. The Gospels say Herod imprisoned John because John condemned his marriage to his brother's wife (Mark 6:17 to 18). Josephus says Herod feared John's influence over the crowds. Both can be true at once: a king angered by a prophet's rebuke, and afraid of how many people listened to him.
What matters is that a Jewish historian writing in Rome, sixty years later, independently confirms the man, his baptism, his reputation, his imprisonment and his death under Herod.
Why this is a big deal
John the Baptist isn't a character who only exists inside the Bible. He was famous enough that a Jewish historian, writing for a Roman audience with no interest in promoting Christianity, recorded who he was, what he preached and how Herod killed him.
And earlier in the same book, the same Josephus names Jesus too.