Anthropomorphism, defined, is "the attribution of human characteristics or behavior to a god, animal, or object". So OK, why should we care about it?
It's super easy to accidentally anthropomorphize our horses. After all, we love them like pets or even children sometimes. They are incredibly sensitive animals and by forming emotional bonds with them, we feel like we can read into their emotions (and that they can read into ours). And to a degree, that is true. But we make a mistake if we forget that horses are horses, with brains different from ours. Just because we feel bonded to them does not mean that they think the way we do or see things from a similar perspective.
This distinction matters because anthropomorphism goes both ways: if we are trying to be extra kind to our horses in a way that we would like, we might overfeed them treats that aren't necessarily good for them. In our attempt to acknowledge their sensitivity, we might assign too much responsibility to them for carrying our emotions and then resent them if they don't act according to our expectations. And if we believe that misbehavior is always intentional and we liken it to the bratty-ness of children, then we fail to see where perhaps it's our own errors in training that are producing those behaviors because we stop short at blaming the horse. Fair equitation means understanding both the abilities and the limits of our horses' minds and bodies.