People were afraid of "it". "It" stands in for the object of the sentence, which is "this strange and terrible beast living in a deep, dark labyrinth on the island of Crete." The blank cannot be filled with "lived" because there is already a verb in the sentence. As Mr. Kimball said "that lived" is normal because "that" opens a relative clause, which acts more like an adjective describing the beast.

It could be "that lived", but not just "lived." It could also be "that used to live" or "living." The first two are verb forms in relative clauses, but with "living", you're using the Present Participle as part of a participle phrase that functions as an adjective to define "beast."

because he existed in the cave _continuously_, not for an instant, but for an indefinite time, i.e., you need the continuous tense, "living."

The verb "living" needs to be in the present participle form to agree with the subject "beast".