You could easily write a 100 page essay on this topic. Religion doesn't fully cover it, since non-abrahamists and atheists mostly still hold the same vitriol as everyone else does. It's helpful to listen to what reasons the haters have to raise about it themselves, since it's easy to extrapolate from there. Some commonly encountered reasons:
"It's revolting" <- acquired reflexive behavior towards an act the society already deems unacceptable, essentially a circular, self-reinforcing taboo enforced by peer pressure; individuals typically consider the matter "settled" (since they don't hear of anybody questioning it - well no wonder, when the media they listen to are censoring it to begin with) and are hostile towards people trying to question it because "Pandora's box" or whatever.
"It's abuse" <- well, we have to admit that real abuse does exist, but this is about the cases where it's applied on a blanket basis. The general negative image of zoos perpetrated in media and in science carries the biggest blame for this. It's essentially an appeal to science; science (or rather, pop-sci) is what perpetuates the stereotypical imagery of animal behavior that many people clueless about animals subsequently adopt. Hence the people who resort to this most eagerly are be the "socially conscious" highly urbanized college Starbucks types. The follies of modern science practice become extremely apparent once this point gets further dissected in depth, but that's a tale for another time.
"It's sick/Only sick and dangerous people would do it" <- much like above, but the focus is shifted on media image rather than science. And also poor understanding of psychology and psychiatry - which might be one of the most socially destructive forces in the world today in general. Again there's a conformist element to it, but this time it's a peer applying pressure to an outsider and trying to absorb him into the group. Something to specifically highlight here is the cursed label itself: "zoophilia". Note that in normalspeak, the suffix "-philia" already became colloquially understood as a shorthand for "sick fuck". Thus a key step to dispelling the stigma will be to rebrand zoophilia under a new name without the negative history of centuries of media systematically crusading against it (for ultimately unrelated reasons).
"It's unnatural" <- usually uttered by not very bright people who also drive a fully natural car and own a cat raised on fully natural processed kibble.
"It debases the animal/humanity" <- the former is weird in general, and probably arises from some esoteric, magical perception of nature to begin with; the person would probably reveal themselves as rather loony by society's standards as well if further probed about the matter (think "crystals and chakras"). The latter is classic human exceptionalism in practice, though it begs the question why the "superior omnipotent human" should restrict himself from exercising his "dominant power" over his "inferiors" (this is sensitive ground; bad handling of this might end up inadvertently apologizing genuine abuse).
"It's primitive" <- a point probably reserved mostly for developing countries; zoophilia perceived as a symbolic act of socioeconomic failure. Rich and mighty westerners have outlawed it; rich and mighty westerners are rich; hence we should follow western mores to the letter and we'll be rich too, justification be damned. But even in countries like Japan, anti-zoo sentiment seems to correlate with the level of westernization of their society. Zoophilia is an act with extremely bad PR on an international scale (since economically relevant countries are mostly ensnared by the above points), and thus many people will want to sweep it under the rug just to not look like an indecent country.
I'd say the main problem of all these is that people in general have become too urbanized and too detached from the reality of interacting with animals. They are, in short, stupid about animals. They have a dog and suddenly they are Dunning Kruger experts on what a cow behaves like as well, and then they apply the same myopic set of morals to the entire animal kingdom simultaneously. But weirdly, rural folk tend to be stupid about animals as well, if they only perceive them as a money machine and never bother interacting with them in their off hours. If you take your donkey for granted and don't cherish him, if you only see him as a meat tractor, you keep overworking him, keep him in poor conditions, never check on him unless he's lying stiff, then no wonder you think he's just a dumb beast of burden incapable of and unworthy of love and affection.
