As @Todd noted, the episode was written by Dorothy Fontana. She wrote under the pen name DC Fontana to disguise her gender. As I noted in my review of Far Beyond the Stars, the character of Kay Eaton, played by Nana Visitor was based primarily on Fontana.
Yesteryear was the only TAS episode Fontana wrote, but she was brought aboard TNG quite early in its production, and she co-wrote Encounter at Farpoint (with Gene Roddenberry, and they earned a Hugo nomination) as well as 4 other episodes. She also served as an associate producer for the first half of series 1 until she and David Gerrold quit the show due to the interference of Roddenberry's lawyer Leonard Mazlisch. William Shatner's documentary Chaos on the Bridge is well worth checking out on this front, and it's on Netflix.
Fontana would only return to Trek on TV once after leaving TNG - she co-wrote an episode of DS9, but she also wrote a number of computer games. Outside of Trek, she wrote for a wide number of TV shows, including Babylon 5, Earth: Final Conflict, and Beast Wars: Transformers.
In terms of connections back to the original series, most of the TV cast were in TAS, with Walter Koenig being the only actor to not return, even though he did write an episode. James Doohan was noted for his vocal talents and so voiced a wide number of characters, as did Majel Barret - as well as Scotty and Nurse Chapel, the pair voiced new regular cast members Arex and M'Ress, and many recurring characters. In Yesteryear, Doohan plays no fewer than seven different characters, including Scotty. Even though Chapel and M'Ress don't appear in the episode, Barret voices all the female characters in the episode - including Amanda Greyson, who was played by Jane Wyatt in TOS and the movies.
Of course, Mark Lenard returned to voice Sarek. He's one of three actors outside the main cast to reprise a TOS role in TAS (the other two being Harry Mudd and Cyrano Jones' actors).
Spock talked of having a sehlat as a pet in the TOS episode Journey to Babel (also written by Fontana), and the description given in that episode (a fat teddy bear) was one of the starting points for the character design seen here. A sehlat would later be seen in Enterprise with a less cartoony design
Even when TAS wasn't considered canon, this episode generally was, with elements such as Amanda's surname, the location Vulcan's forge, and some other place names showing up in other Trek series.
I-chaya's temperment and character was based somewhat on Fontana's cat at the time. She pushed quite heavily to keep the euthanasia element of the story in place. NBC were against it, but the deal with Filmation said they had final approval of story elements and they sided with Roddenberry and Fontana
This episode was part of the first recording session, which was lauded in the press as the return of the Enterprise crew. Even though everyone's part was recorded separately, an effort was made to get the main cast in the studio on the first day.
Billy Simpson, who played young Spock read all his lines in his audition. The production staff decided he was the best actor who went for the role, and used his audition tape for the audio in the episode rather than have him come in and record his lines again. This is partially why he pronounces I-chaya differently to the other characters, and that there are some odd inflections in his line readings.
This is the second episode of TAS, but aired first in California as George Takei was running for District Councilman at the time, and if an episode he appeared in aired in that timeframe it would have counted as an unfair allotment airtime for him even though he wasn't putting his politics forward
As to my thoughts
It's very interesting to see classic Trek in a half-hour format. There's very little padding here - we come in at the end of the Enterprise's mission to observe history, and the change to the timeline is spelled out almost immediately. There's no time for any complications here, nobody questions Spock's general existence or Kirk's memories of him, and Spock's in Vulcan's past before the second act. Even then the plot moves fast to get everything done in the run time.
And the reason I said that this is Trek in the half-hour format is because that's exactly what it is. The filming media doesn't matter, or even that the original cast is mostly back. The story is pure Trek from start to finish. The SF high concept - history is inadvertently altered and Spock has to go back and save himself, as he remembers already having happened - is just a framing device for the more powerful persona drama between Spock and Sarek, Spock's dual nature as half-human and half-vulcan, saying goodbye to a childhood companion, and coming of age as a result. There's so much to unpack here, one could easily write a lengthy article.
The story is so good that it's easy to overlook or ignore the technical limitations. Doohan was the only veteran voiceover artist and it shows, with DeForrest Kelly's line reads feeling disinterested, and while Shatner and Nimoy's are decent, they feel isolated, possibly as a result of limitations of the equipment used, or that it was part of the first session.
The animation is pretty cheap as well, but that's Filmation for you. The original elements to the show - the alien designs and landscape are very Filmation. I tend to think of this and the 1979 Flash Gordon cartoon together, even though there's 5 years between the end of TAS and the start of Flash Gordon because of the similarities in their look and feel (and they tended to both be in blocs of kids' TV programming when I was 12 or so). There's a lot to be lauded in the design work - we've got a non-humanoid alien as a member of the Federation science team (and of course Arex and M'Ress in the regular cast), and it would have been very expensive to portray as much of Vulcan as we see in the episode in live-action.
A lot of people have said that this episode is good enough to have been filmed as live-action, but I think that somewhat misses the point. TAS is as much Trek as any of the other TV series and movies, and the story is good enough to be Trek. It doesn't matter that it's a cartoon. Infinite Diversity in Infinite Combinations.