The best way to understand a group like that is to experience it first-hand, though you could always read information about Otherkin such as this and that and {insert word that comes after a 'this and that' for a third thing}.

Basically, Otherkin are people that feel deep down that they are of a different species, common ones being dragon, fae, and elf. The ones that correspond to 'normal' animals (e.g. wolf) are usually called Therians.

The process called 'awakening' is when someone realizes they're Otherkin and what type of Otherkin they are. It's usually a slow process however.