In Log Horizon, Meow's burger meat come from a real animal, but when the animal is killed, why doesn't its body turn into particles and disappear like other monsters?
It's true that the meat or other ingredients used in food are obtained by killing monsters. I must clear the assumption that the monsters don't turn into spirit energy and disappear.
They do disappear.
The meat that is used for cooking is actually an item drop from those monsters. Which means that only the monsters that drop such items can be used for cooking. In the first season of Log Horizon for instance, they started hunting Imperial Forest Boars for food. The meat obtained from that is actually an item drop. However, the wiki also says that a chef class player can also harvest the monster for more meat but how exactly is it done is not explained. It might be possible that the monster just drops more meat when a chef class player kills it.