The correct answer is (E).
There first part of this sentence, the independent clause, tells us that the professor's lectures were "uninteresting," while the dependent clause (which begins after the comma) gives us the big clue that these lectures caused people to fall asleep. What kind of effect did his lectures have? One that made students sleepy, so the second blank will have a definition that matches this idea. The first blank is a bit harder to predict right now, so let's try to eliminate some answers based on the words for the second blank.
(A) "esoteric" means "understood by only a few," so this doesn't match the sentence (remember, we need a word that means "causing to fall asleep"). (B) "hypnotic" is promising, so don't eliminate this. If the lectures were (C) "captivating," or really interesting, they certainly wouldn't make people sleep, so eliminate it. (D) "burgeoning" ("growing, flourishing") doesn't really make sense, so eliminate this as well. (E) "soporific" means "lethargic," so you should also keep this choice. As for the first blank, we are left with (B) "energetic" and (E) "monotone" ("uniform"). Which would put students to sleep? An energetic lecture would wake them up, so the best answer is definitely (E).
