The debt load of veterinarians is pretty staggering. Going to vet school is as expensive as going to medical school, but the pay is considerably less than an MD makes. It is not at all unusual for a vet to graduate with $300k+ in student debt, and to still be paying that debt down well into their fifties or even sixties.
You also have to deal with pet owners who are... let's say irrational. People seem to be even more willing to believe in homeopathy or nutritional fads or essential oils when dealing with their pet's health than with their own. The bills for veterinary care are not as high as for human medicine, except that most people don't have pet insurance and so pay much higher out of pocket bills than for their own health care, which makes the emotional stakes even higher. And owners will not hesitate to trash you on social media. The issue here is that to a much greater extent than doctors, veterinarians gain new patients by word of mouth and recommendations. So reputation is even more important than it is to doctors. A few ugly, well-placed reviews in Yelp or Facebook can do considerable harm.
All of this leads to an extremely stressful job, especially for younger vets or vets with young children.
Especially if they owners simply can't afford to pay to keep the animal alive.
Imagine being a surgeon and telling someone that their child needs their appendix removed, and the parent sees the cost and says "eh, let's just put them down".
The linked article goes through a list of qualitative factors which could be linked (though it does not seem to attempt to quantify their correlation).
I could easily imagine that more depressed people pursue veterinary jobs because being around animals makes them less depressed.