A thought just struck me that from Back to the Future II onwards, Doc was quite adamant of the need to destroy the DeLorean, not because it was a bad car or anything, but because the risk of time travel was too dangerous. Yet, at the end of Back to the Future III, we see Doc has created another time machine in the form of a train!
Why did Doc create another time machine after being set on destroying the DeLorean because of the dangers of time travel?
Although it's never really discussed in the movies, there's really one big event to convince him to destroy the DeLorean.
Namely, a bumbling aged idiot named Biff was able to figure out how to steal it and work it inside a day, destroying everything Doc had in the process. It's easy to use nature was it's danger - power like that never should have slipped out of Doc's hands.
I agree with comments and the answer from Charles that a quieter life and family gave him the distance and time to rethink it. He had really invented a wondrous thing - properly controlled, and kept safe in the right hands, it could be done right. My own thinking is that this led to his new design. It's a lot harder to steal a train, he had experience with that. We never see the inside of the thing, but I imagine a steampunkish design with physical locks that only he would know how to operate. It would be a far cry from a hop in and go vehicle that a car represents.
I agree with "you can't keep a good scientist down". The DeLorean time travel experience was a continuous, rolling catastrophe that required full time focus from Doc and Marty, so it was no wonder Doc had a meltdown or two about time travel. With the advent of Clara and his two sons, and a little bucolic quietude in his life for awhile, Doc dreamed of trying it again - and this time he could get replacement parts - or make them - in his new 1885 home. Also, he now had a nice big cockpit for the rig, so he could show his whole family the wonders of the past and future. (Since the new steam machine could fly, no tracks necessary in the Jurassic)
There is a very important reason that’s never addressed in the movie and should have: by staying in the 1800s doc is changing history. Even worse, Clara was dead in the original timeline and their kids obviously weren’t going to exist. Therefore, relocating to a date after 1985 is actually the correct thing to do to preserve the time continuum.
I’m actually a bit annoyed that they point out in the movie that Clara should have died and Doc doesn’t state that then she HAS to come to the future with them.
Canonical information is limited
From the movie we discover Doc's stated reasons.
The gift of the photograph could have been a reason, but it also could have been nothing more than part of the preparations for an already approved/decided-upon journey.
When Marty asked where Doc Brown was going, "back to the future?" the Doc specifically states that he's not returning to the future. "No, already been there." Curiously, he's already been to the past, too. He doesn't tell us where he's going, but he travels through time to get there. That's the closing scene of the trilogy.
There are potential story writing reasons...
There are a number of potential story writing reasons why the time train exists. None of these reasons have anything to do with the story itself or the world created as the framework of the story.
The time train certainly serves a story purpose by allowing for a western-style epilogue where everybody says their good-byes and the move ends on a distinct up-beat. There can easily be no other reason for the time train than this.
The time train may symbolically represent the idea that the "journey never ends," which is a big deal when it comes to time travel and its many potential ramifications. It's very common for aspects of a story (including its props) to serve symbolic or metaphorical purposes.
Robert Zemeckis may have wanted nothing more than to do one more really cool thing. And I have to admit, it would have been a riot working on the crew that designed and built that train.
Zemeckis may have wanted nothing more than to end the movie the way the first two ended: with an open-ended cliff-hanger. Doc, sailing off into the sunset of time....
Finally, Zemeckis may have wanted nothing more than to end the movie with the signature visualization that had come to represent the entire franchise (and was likely intended to do just that). In this case, the train is nothing more than a way to reflect an 1800s Delorean for a unique movie ending.
But if you're looking for in-world reasons...
But if you're looking for an in-world reason to explain why the time train exists, all I can tell you is that sometimes the quest for answers doesn't prove fruitful. Worse, it's often a huge mistake to insist that every aspect of fiction have an in-world (or, worse, a Real World) explanation.
The several opinions that have been proffered are interesting — but without commentary from Zemeckis and/or his writers, they're just speculation, and pretty wild speculation at that. I have two words for you: prime directive. Showing up in ancient Rome with a time train would cause its own problems with history and it's a bit harder to disguise or hide than a Delorean.
I have a sneaking suspicion that Zemeckis didn't have an in-world reason for the train beyond what Doc Brown said, which was really nothing more than a rationalization to produce the epilogue Zemeckis wanted.
But I'll defer to additional canon, if such can be found.
