Let's begin this monologue with an observation about myself: I wouldn't say I have a personality that is instantly endearing to pets. When a dog or cat comes up to sniff me for the first time, I could just be any human. There is no instant connection, there's nothing special about my demeanor or identity that draws animals to me.
But in my experience, after I've spent time with animals, I form pretty strong friendships with them. I consider pets to be family, and I would say that pets of mine have all grown to consider me either a friend or a "family" member of sorts. People are impressed by how much pets like me.
I assure you that it is not a reflection of anything impressive about me. I am not special. I simply allow animals to interact with me on their terms. I respect them, and I listen to them, and they ultimately appreciate that. So, I'm well-liked by nonhumans who I've spent a lot of time with. Especially if we're talking about cats. Cats like me more than they like most people. Dogs are naturally social with humans, but cats have to build those social bonds from scratch.
I believe I understand how to interact with animals. I'd like to share specific heuristics for doing so, rules of thumb for what to do to understand your pet. As subtle as it is, I think I've done enough introspection about it, and I have enough education in cognitive science that I think I can do a good job explaining this.
So, if you want to "talk to animals", then this is your guide from a first generation immigrant who's slowly picking up the language. I'll discuss first how to listen appropriately, then how to convey what's on your mind.