I think animals rely on our facial expressions, tone of voice, heart and respiration rate, smells and body language, and I think they've learned to do this as we're part of their pack or herd so to speak.
So rather than a 6th sense they 'read' us in far greater detail than we realise, and probably read and understand things we don't even realise we do/say.
I went through a tough time recently and spent some time with me horse and his field companion. He has never been a horse that likes to be cuddled or petted a lot, but I think he's learned that certain things he does make me laugh and that gets him more of my attention which is desirable as I'm the food provider, he started playing with my zip and took my hat off and when that didn't work, wandered off. The female horse stood over me dozing when I sat on the floor - like she would a foal or other vulnerable horse, I am part of her heard and usually I'm the provider and protector, but at that time I was vulnerable and from my body language and demeanor she understood that and took the 'protection' role, even though she knows I'm not a horse.
I sometimes think that humans could take a lesson on inclusivity from animals - they know we're not the same species as them, yet accept us anyway on face value as part of their group, once they know we can be trusted.
They also have an uncanny sense of time, I finish work at 6, and feed them around 6.15, they'd always be at the gate waiting at 6.15, despite there being about 4 months of the year that it's dark at 6.15 and then varying degrees of daylight after that. If I was early and got there at 6 they were halfway down the field, working their way down. They didn't hear my car because I walked, and I know the point they can recognise my footsteps from because they whinny when they know I'm there. I can only think they have a much better sense of the light and changes from night to day than we do!