Hello there !
I suppose this means "I can read you like an open book", but what is this dog doing here ?
Yes, that is what it means. The adjective 'dog-eared' refers to the pages of a book being turned over at the corners due to frequent use - so they look like a dog's ears. It is generally used to mean 'worn' or 'shabby', but here it is just an emphatic exclamation, like 'goddamn' - it has no real meaning.