To understand how to figure misplaced / dangling modifiers, we must first understand the message that the sentence is trying to convey to us. So let us take a look at the answer choices:
A. The veterinarian, found in the woods with a broken leg, offered to help the dog.
Was the veterinarian found in the woods with a broken leg? And if so, by whom? And if they had a broken leg, why are they concerned with helping the dog, who seems to be fine? I do not think that answer A is the right choice, so let's move on.
B. Found in the woods, the veterinarian offered to help the dog with a broken leg.
Aha, at least now the dog has something that requires the help of the veterinarian. But why is the veterinarian being found in the woods? Is he some sort of hermit? Probably not. Moving on!
C. Found in the woods with a broken leg, the veterinarian offered to help the dog.
Now, we are completely stumped as to what was found in the woods with a broken leg. Because it's unclear, we can dismiss it as an option choice.
D. The veterinarian offered to help the dog found in the woods with a broken leg.
NOW we're onto something! The dog was found in the woods with a broken leg, and the veterinarian offered to help. From the looks of things, this makes the most sense, and therefore, option D is the correct answer.
