There are many humans and animals that are able to comprehend things on a vaster level than just via sound/inflection/pattern. I'm not exactly sure where this communicative ability comes from, but it's there. We humans generally refer to it as a "bond"... but no matter what it is, it's totally awesome to have and definitely something worth trying to perfect.
I believe animals can "sense" who can understand them and in what ways, (I believe we humans can do the same thing, but I think 90% of the time it's totally unintentional with us)- and I think that ability in and of itself is how many of us end up selecting a pet (or their selecting us), who ends up working as our familiar.
On a psychological level, animals communicate through visuals and smells.
I've tried many times to send visuals to other animals and have observed that some hear me and some don't. In this way, animals are just like people- some are able to accept these ideas and some are unable/vehemently opposed to the idea of their being any "link" to animals, humans, or the spirit realms. To each their own- if that human or animal is supposed to get to a point where they can acknowledge one another on a deeper level, they will-- in their own time.