Is "puzzled why" correct as well? Is it just a choice of style? Or is there a difference in meaning? And if so, I'm curious (as to) what that difference is.
Grammatical rules that apply to units larger than a word apply only to constituents.
The psych predicate adjective (be) puzzled can take a number of complements, including tensed that- and wh-clauses
and a number of other constructions, often with prepositions
There is also a Passive construction (like most psych predicates, which often come originally from passive participles), which allows a by-agent.
As to the as to, that's just a longer (and optional) compound preposition governed by puzzled, like at and about.