During the time of the Aḥaronim, the custom spread in many places not to slaughter animals on Yom Tov, for two main reasons: first, there was concern that people would do business with the meat, and second, many animals proved to be non-kosher. We will now explain.
In the past, most Jews lived in rural areas, raised livestock, and knew how to slaughter them to feed their families. On Yom Tov, several neighbors might join together to slaughter a lamb, taking care not to talk about price or weight. Rather, each neighbor would remember what part he took, and after Yom Tov, he would calculate the cost of his part and pay the animal’s owner (Beitza 27b; SA 500:1; Peninei Halakha: Shabbat 22:3). Later, during the time of the Aḥaronim, slaughtering became a specialty. A butcher would buy animals and slaughter them for the whole city. On Yom Tov, butchers needed to slaughter many animals and distribute the meat to many people. Since they would be unable to remember after Yom Tov who took what, they were likely to end up doing business on Yom Tov, which is prohibited by Torah law (see Peninei Halakha: Shabbat 22:2-3).
The second problem was that in many places adhesions on the lungs were common, to the point that sometimes half of the animals slaughtered turned out to be unkosher. It is true that many opinions maintain that even under such circumstances, slaughtering on Yom Tov is permitted (Rambam; Rosh; Rashba; SA 498:8), but others are stringent, since there is a good chance that any given animal will turn out to be unkosher, meaning that, retroactively, the slaughtering did not provide food for Yom Tov (Or Zaru’a; Ra’ah; Ran).
Therefore, many Aḥaronim ruled that livestock should not be slaughtered on Yom Tov. Fowl, though, was still slaughtered on Yom Tov. Since they are small, the meat did not need to be divided among many families, and there was less concern that people would end up doing business. Moreover, the percentage of birds that turn out to be unkosher is much lower than that of animals (MA 498:16; SAH 16; MB 498:49; Ru’aḥ Ḥayim 497:2).
Nowadays, when it is easy to keep meat refrigerated, the general practice is not to slaughter at all on Yom Tov, in order to avoid all the effort of slaughtering, checking the internal organs, skinning, and salting. Nevertheless, in cases of great need, one may still perform sheḥita on Yom Tov. Thus, if an animal is close to death, one may slaughter it on Yom Tov to avoid losing all the meat (as an animal that dies without sheḥita is not kosher), on condition that there is enough time to roast a kezayit of the meat and eat it on Yom Tov (SA 498:6; MA 16; Ḥayei Adam 89:6) A dairy cow may also be slaughtered in this case (as explained below, 6:6).