Much of what occurs in this final episode of the "Star Trek" violates what you'll later see in series such as "Star Trek: Enterprise" as well as the movie "Star Trek: First Contact". For example, the first captain of Enterprise is Captain April--and they also say this same ship, NCC-1701, is the first Federation ship with warp technology. My advice is to just ignore all this....get over it and just watch. What you'll see is a reasonably interesting but, once again, horribly animated show.
The Enterprise is transporting Mr. April and his wife. April had been the first commander of Enterprise and his wife had served as the ship's doctor. They're old and awaiting retirement. However, en route to their 'decommissioning' ceremony, the ship is sucked into an opposite universe--one where folks are born old and progress to infancy! And so, the entire crew is growing younger and younger and it's up to April to command the ship back to our reality. Quite the coincidence that he happened to be aboard, huh?!
This is not a brilliant episode by any SANE standard. But it is interesting and despite it's excremental animation, it's a decent place to end the "Star Trek" animated show.