The gemara says that although generally it is prohibited to prepare on a Shabbos or Yom Tov for afterwards, even if not melacha is being violated, as the mishna 113a implies, there was an exemption made for קניבת הירק in the late afternoon of Yom Kippur. There is a machlokes what exactly קניבת ירק means. Rashi says it refers to the removal of the leaves to prepare the vegetables to be cut and sliced after the fast. Rashi seems to hold that the heter was only to begin the preparation but not finish the process. The Ran doesn't agree with Rashi because the removal of the leaves is considered borer, and therefore explains that it refers to the washing of the vegetables which would normally be a problem of mesakein, but is permitted on Y.K. (this approach actually fits better with rashi 113a regarding the folding of keilim where rashi seems to hold the nature of the issur is fixing, not doing a tircha on shabbos and y.t. for afterward). Tosafos implies that we are referring to the actual slicing and cutting of the vegetables and is therefore compelled to distinguish this case from the cases earlier in shabbos where there is a melacha involved in the cutting of vegetables. According to Tosafos we should permit any preparation for after the fast, and not limit it to just starting an act of preparation. The Shulchan Aruch 611:2 writes that technically kenivas hayerek is permitted but nowadays the minhag is to forbid it as we see from the end of the gemara that people started to be lax and being the preparations prior to mincha.
Even at the time when it was permitted, it was only permitted after the time of mincha ketana. The gemara says that the rationale to permit it was עגמת נפש. Rashi explains that since the entire nature of Y.K. is to be a day of inuy, they permitted קניבת ירק since being involved with food when you are very hungry compliments the inuy and is conducive to the nature of the day. It is for this reason that it was only permitted late in the day when people are hungry and enticed by the food that they are working on. However, the Ba'al Hameor (printed in the inner margin of the gemara) explains that the agmas nefesh that the gemara is referring to does not take place on Y.K. itself, rather, if they would not permit preparations on Y.K. there will be too great of an agmas nefesh after the fast ends during the time it takes to prepare all the food. They permitted preparations on Y.K. to avoid עגמת נפש. Based on the ba'al hameor, the agmas nefesh is only a justification for the heter, but doesn't explain why it was only permitted starting from mincha and onward. To deal with this the Ba'al Hameor creates a new prohibition, a concern that one may come to eat from the food that they are working with. This would be a concern all Y.K. long until the very end of the day where the fast is imminently over so that the person has פת בסלו and won't come to eat from the food.
The Ba'al Hameor seems to create a new prohibition that we don't have a source for in the gemara. It would be prohibited to busy oneself with food preparation out of fear that they may come to eat from the food they are woking on. We find a similar prohibition regarding chometz in Pesachim 10b regarding whether one should do a bedika on Pesach (if they didn't do one before). The gemara suggests that it may be best not to do a bedika because one may come to eat from the food. Similarly, Tosafos 2a explains that it is for this reason that chazal imposed bedika on top of bitul, because with bitul alone we are afraid that you will come to eat from the food. The gemara 11a explains that by other issurim we don't find such a prohibition because one naturally stays away from it, whereas chometz which is permitted the entire year, one will not be as careful to stay away from. A similar reasoning would apply to Y.K. that one has a tendency to not keep their guard up so we are afraid of him coming to eat from it.
Based on this, it should prohibited on Y.K. to prepare meals for children until late in the day. Yet, the Rama 612:10 quotes from the Terumas HaDeshen that it is permitted to touch food and drink on Y.K. to feed to children and we are not afraid that he will come to eat or drink it. The M.B. explains that on Pesach where he is eating other foods, we are afraid that if he is involved with chometz he will eat that too, but on Y.K. where one is abstaining from all foods, we are not afraid that by working on preparing food he will come to eat it. Although the Sha'arei Teshuva 3 cites the har karmel (21) who says it is still better not to be busy with food prep, it sounds like just a chumra, whereas according to the ba'al hameor it should actually be an issur.