What did Christ mean when he referred to John the Baptist as Elijah? (Matthew 17:11,12 – Mount of Transfiguration) Is this evidence that he recognized reincarnation as a truth?

The theory of reincarnation is nowhere taught in the Scriptures. Our Lord Jesus and the apostles never, in all their teachings, said a word about reincarnation. As to the meaning of the Savior’s words referred to in the question, we recognize that He was calling to mind the prophecy of Malachi 4:5,6. 

John the Baptist was one of the greatest of the prophets. It was his mission to prepare the way of the Messiah, the Christ, by performing a reformation work. He preached repentance and baptism for the remission of sins and declared the kingdom of heaven was at hand. 

This message was to the Jews and to them only as they had been the Lord’s chosen people for many centuries, but a new dispensation, or time period, was about to happen, and it was John’s mission to prepare the people for the new order of things. This he did in the power and spirit of Elijah (Luke 1:16,17). We know positively that John was not Elijah for we have his own testimony on the subject: “And they asked him, What then? Art thou Elijah? And he saith, I am not. Art thou that Prophet? And he answered, No” (John 1:21).

Additional Resources: 
Christian Questions Podcast
Episode #1148: “Ghosts, Reincarnation and Humanity – What’s Real? (Part I)”
Examining the biblical feasibility of ghosts being disembodied souls
Preview Video
CQ Rewind Show Notes

Episode #1149: “Ghosts, Reincarnation and Humanity – What’s Real? (Part II)”
Shining the light of Scripture on reincarnation and mystical practices
Preview Video
CQ Rewind Show Notes