Answer: The right answer is the B) Having read your letter, my German shepherd will stay inside after dark.
Explanation: Just to elaborate further on the answer, it can also be added that modifiers aim at clarifying or giving more detail about an idea. "Having read your letter" indicates an action, but there is no reference to the person who did it, or to the doer. That doer should have been the subject of the main clause that follows, but that has not been the case. Instead, it seems that the German shepherd, which is the subject of the main clause, was the one that read the letter. By neglecting to add the pronoun I in the main clause, the participial phrase "having read your letter" has turned into a dangling modifier.