In Buddha's time, the prohibition was against killing, and not eating meat, and even monks were allowed to eat meat donated to them as alms. That's still how they do things in Tibet, where a small ethnic group of Muslims do all the animal slaughter, for meat which pretty much all Buddhists there eat. It was only with the arrival of Buddhism to the Confucian-influenced world, that monasteries had to be understood as a family making household decisions including about food, and so not killing had to mean not eating meat. For Buddhists the issue is the karmic consequences of being a killer, conditioning your mind like that, and increasing the number of beings out that wish you harm (from their next lives). I hear a converse point made in recent times, maybe people shouldn't be allowed to eat an animal they aren't willing to kill, because they are morally ignoring the consequences of their actions - in some ways it could be better to fully appreciate the animals sacrifice and consume it mindfully, that cause others to kill, and eat unmindfully.
If you look at the end of bull and bear and cockerel fighting, the cruelty to the animals was a concern, but the bigger one was about what impacts spectating had on audiences. We can see this with the professionalisation of boxing with the introduction of 'Queensbury rules', to eliminate facial bleeding, which previously had been the signal to begin betting on a victor. The boxers wellbeing wasn't the concern, but the blood. Opposing animal suffering says something about that person, as well as about animals.
Meat eaters that don't rely on religious law, often make a capacities-based argument, to explain why eating humans is bad, but animals is ok. This is confounded by the fact that pigs, giving the world's second most consumed meat type, are among the most intelligent animals, and so are octopuses, also widely eaten. Some animals like dolphins and whales have been given extra rights based on capacities, in most jurisdictions. Pigs are a good example of our real thinking, because although they have been shown to spontenously employ tool use, they don't make particularly good pets. Whereas, at least in England, there has been frequent outrage about eating horses, which make great pets. This points towards how our enpathy for specific species is a major factor. Eating dogs is widely regarded with horror in the West, but mostly although commercial trade in dog meat is not legal, eating dogs is - eg see moves to ban it in UK parliament. The animals we feel empathy towards, we don't just want killed humanely, we don't want killed at all except to prevent suffering.
The capacities argument can go that animals have no concept of mortality, and that they are very similar to each other so the world is not deprived of a unique voice like by the murder of a human. Our gigantically increased concern for animals on the cute and fluffy end of the spectrum, especially that interact well enough with us to be pets, shows how in practice our views on killing and meat are not generally reasoned, but felt.
I find it amusing the vitriol people have in response to this being pointed out, and the mental gymnastics they do about the various edge cases. I would advocate any defender of meat eating clearly account for exactly which meats should be eaten, seperately to just going with the flow of their local culture and refusing to introspect.
Edited to add I am kind of fascinated by the difference between act and commissioning an. I think the attitude to executioners in Ancient India as untouchables despite provision for the death penalty in Buddhist statecraft, attitudes to the Yamada clan who performed executions for the Tokugawa Shogunate in Japan, or attitudes to other executioners like Jack Ketch for English king Charles II. I feel like there has been a shift in moral reasoning from a focus on act, to one on intention, and societies have generally lost the idea of religious or ritual purity and specific acts being taboo regardless of reasons. It is interesting to note Eden and Heaven are described in the Bible as vegetarian, associating meat-eating implicitly with sin and suffering, and the drives that go with them.
Edited again Peter Singer argues that the expanding of our 'circle of concern', towards having more empathy and respect for others, is the direction of moral progress. In this picture, just as gladiator fights between slaves are just not ok anywhere anymore, that in societies that continue to develop morally, it is inevitable that killing and eating animals will just come to not seem ok anymore, like child labor, or businesses allowing worker deaths purely to save money. Eating animals just will gradually cease to seem a civilised thing - being able to make perfect far cheaper replicas using designed proteins would help of course. I mention this in response to perspectice in @AmeetSharma's post, where the implicit assumption is that it will never be possible to persuade a majority of killing animals is bad. Society has changed in major ways before. Ending the death penalty involved having the 'luxury' of being able to pay as communities for imprisonment, for instance, that just wasn't practicable for subsistence farming societies. Technological, social, and moral change, have often been interwoven.