Watashi wa is correct... but this book has been written in 1905 and the author chose the "ancient" form "wagahai wa".
As for "desu" it's a bit difficult to explain but you must know that "aru" is the verb meaning "to be" while "desu" hasn't a real own meaning and the most common meaning that it takes is "to be" but it can subsitute any verb, when it's clear in the context that you're talking of a determinate verb, for example because you used that verb in the previous sentence...
For example, if you are asked: "Nani o tabemasuka" = What are you eating?" you can answer with "desu" instead of "tabemasu": "Sashimi desu" = I'm eating sashimi; as in that context "desu" is meaning "to eat".
Therefore, to mean exactly "I AM a cat" the author used the verb "aru" whose meaning is "to be" in order to avoid to be misunderstood (as "I am a cat" is a strange thing to say...).
Oops, forgotten about "de" which is a particle with many different applications... in this case it is used to indicate the "nominal predicate" (don't know if you call it this way) with the verb "aru"...
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