One way is to write them just like humans. This is a good device to use when writing a story with a moral. Good examples are the Grimm fairy tales. Some scholars think that the talking animals were simply a device used to portray the worst of human criminals. They aren't really animals at this point, but analogies.
Another way is to represent them speaking with a verbal language but give them distinctly different points of focus. This is a good way to deal with the idea that these are animals without having to dig into a level of realism that can be off-putting to readers seeking an enjoying read. An example of this is the seagulls in "Finding Nemo".
The third main way is to write the perspective of the animal with the best understanding of the species' psychology avaliable at the time. This is quite difficult as a writer for two reasons. The first is that it requires a lot of research. The second is that without dialogue it is very difficult to "show not tell" for a character. A great example of this is "White Fang" or "The Call of the Wild."