The verb of the sentence should be in present continuous tense as it implies that an action is taking place at the time of speaking. Thus, option C "realizing his mistake" is correct as the verb + ing form of verb is used in the continuous tense.
"His mistake is realized" in option A is grammatically incorrect in this context.
"He realized his mistake" in option B is incorrect as the noun 'the guest' does not need to take the pronoun 'he' immediately in the clause.
"Realized his mistake" in option D is incorrect as the context indicates a subordinate clause. It would have been correct if it were the main clause and the clause "apologized ...." were used after the coordinating conjunction 'and'.
Thus, option C is correct.