So first things first: we need to find the subject of each sentence, and then we need to find what the modifier of that subject is. To find the subject, we need to ask this question: who or what is doing the action here? In other words, who or what owns the sentence's verb? I've underlined the verb:
A) "the plot" is the subject (because it's the thing that "seemed a little thin")
B) "Raul" is the subject (because he's the one who "turned on the tv")
C) "The firefighter" is the subject (because she's the one who "rescuedthe woman")
D) "Jim" is the subject (because he's the one who "spotted an eagle"
Now that we know the key parts of the sentence (the subject and the verb), we can start to pick out the modifiers. A modifier is a group of words that describes something else in the sentence. The thing a modifier describes doesn't have to be the subject, but knowing the subject and verb can help us understand if a modifier is used correctly.
Let's look at A. We know "the plot" is the subject and that the plot already has a verb: "seemed." The modifier is "after reading the whole book," since this is a phrase that wants to tell us more about the person who read the whole book. But that person isn't mentioned in the sentence, so the modifier is accidentally describing "the plot," since "the plot" comes right after the modifier. That means that this sentence is telling us that the plot read the whole book. But plots can't read! So this is a misplaced modifier. To fix it, we'd need to say something like, "After reading the whole book, Billie thought the plot seemed a little thin."
B is fine, because "after he finished dinner" describes Raul, who is definitely the one who finished dinner.
C is tricky, though. Who is the one wearing the blue nightgown? Is it the firefighter or the woman who was rescued by the firefighter? Logic tells us that the rescued woman was wearing a nightgown, but the grammar of the sentence does not tell us that. So C has a misplaced or ambiguous modifier.
Finally, D is fine. We know that "flew over the Grand Canyon" can only descirbe the eagle, since we're told that it flew over the canyon. If it had said "Jim spotted an eagle as he flew over the Grand Canyon," then we wouldn't know, grammatically speaking, if it was the eagle or Jim who was flying.
