How do vets handle animals?

Training. A lot of it is reading body language, but leverage plays a part too. I was a tech for almost ten years and handled literally thousands of animals. There were only 2, an Akita and a Rottweiler, that we couldn't do anything with. Confidence and lack of hesitation are vital. Sometimes you can restrain them quick enough that they can't react. There are a lot of tricks and methods and you eventually develop your own "style".

Restraint is one of the main reasons some procedures are better done in the treatment area away from the owner. Some dogs and cats behave better on their own and some procedures and restraints can worry an owner because they look bad if you don't know what you're seeing.

Experience, training, not showing fear, knowing what to do with difficult animals and putting in place the means of being able to examine any animal! And good vets don't ignore the value of having an experienced OWNER there to control a difficult animal.

They know how to handle animals from working around them all the time. They do not show fear or anger and speak softy to them.

Each one adapts to their own style, everyone has a different way of doing things. They are taught the best ways of restraint and etc.