The overall sentence structure is called a "predicate nominative" because the verb of the main clause is "is". Thus, this structure makes the predicate a restatement of the subject. For ex: "Biden is the President."
"What surprises me" is an independent noun clause whose subject is the relative pronoun, "what", and this entire noun clause is the subject of the sentence. The antecedent of "what" is stated later. Normally, a pronoun's antecedent precedes the pronoun itself. The antecedent of "what" is "that they are fond of snakes and lizards".
"Is" is the verb of the main clause.
"That they are fond of snakes and lizards" is an independent noun clause and the predicate of the main clause, as well as the antecedent of "what". "That" is a subordinating conjunction and introduces the second noun clause. "They" is the subject of this second noun clause. "Are" is the primary verb of this 2nd clause. "Fond" is an adjective that modifies "they" making the 2nd clause's structure a predicate adjective. "Of" is a preposition. "Snakes and lizards" is the (compound) object of the prepositional phrase, "of snakes and lizards" which all together is an adverb that modifies "fond".
This overall sentence structure is not good. This sentence can more simply be written, with the exact same meaning, as:
"That they are fond of snakes and lizards surprises me."