Is it ever explained, in any book or episode, why Flint (from the Original Series episode "Requiem for Methuselah") was immortal? I know he had instant tissue regeneration or something like that, but why?
In short, no, it's not explained in the script, nor has there been any substantial information provided by the cast or crew.
The sole canon source I can locate that discusses the issue (an interview with show's writer Jerome Bixby in Captains' Logs : The Unauthorized Complete Trek Voyages) leads to the conclusion that it was simply a fortunate genetic anomaly:
"I always wanted to do a story about a neanderthal who found himself gifted with immortality, who lived up to the present day. Learning, learning, learning throughout this enormous lifetime, mastering the arts and sciences through philosophy.
In this context, the word 'gifted' would generally refer to a natural talent or ability.
As to the manner of his immortality, McCoy simply hand-waves it away as being an artifact of ...
McCoy : Instant tissue regeneration coupled with some perfect form of biological renewal.
... and that it's somehow connected to the planet Earth ...
McCoy : He's dying. You see, Flint, in leaving Earth with all of its complex fields within which he was formed, sacrificed immortality. He'll live the remainder of a normal life span, then die.
Moving down the canon scale, Flint (now going by the name Dr. Evergreen) appears in the Trek Novel The Rise and Fall of Khan Noonien Singh, Volume 1 where Gary Seven identifies his condition as being a "unique genetic mutation", albeit one that is rare rather than singular:
Seven had his suspicions. “You’re an immortal, aren’t you?”
Now it was Evergreen’s turn to look surprised. His head jerked backwards as he stared at Seven with startled eyes. “How the devil do you know that?”
“It’s the only logical explanation,” Seven replied. Although it was incalculably rare, he had encountered this unique human mutation before. “I assume your injured flesh has already regenerated?”
It's worth noting that the same plot device (an immortal cro-magnon telling his story) was recycled by Bixby for the film "The Man From Earth". Again, absolutely no explanation is offered for how the protagonist has survived since the stone age.