Reading articles will give you the best direction about spirit animal communication - finding.
You'll probably get a summary of a fewer method :
Animal spirits frequently disclose themselves to us, as human beings, via our deep subconscious during sleep and mostly meditation (meditative states). During these periods and sessions, the spirit animals signify precious information about our past, present and future, as well as help us with situations we are dealing with in our lives.
Animals also show themselves during dreams and meditations to exclusively place themselves as our spiritual guide, and thus our animal totem. This consumes us to pay attention, and learn more about what this animal symbolically represents so that we can work with our animal totems.
These were the more common methods as the poster above me has stated.
Some existing methods recommended to learn about are shortly these:
Spirit quest - a special ritual dedicated to asking your spirit animals in, to aid you, guide you, or to simply formally reveal themselves to you and show their presence.
The animal shows your its presence by its own will - sometimes it happens that your guides or your spirit animals are the entrepreneurs of your communication. You can have high expectations for such an experience.
Spiritual journey (spiritual seeking) - spiritual journeys are very common experiences seeking for some unique spiritual achievements including spirit communication, in many cases - spirit animals. Remember to read more about "Spiritual Journeys" and "Shamanic journeys".
As far as you can see I just brought up some further methods and generally defined them, however I don't believe it covered everything that an article could have done. So search for these basic terms, read, and start working on it.