na-adjective
seeming to have a reason; appearing to be for some cause
Archaic adjectival noun describing someone or something that looks like there is a particular reason or circumstance behind it. Not used in modern Japanese.
彼の故ありげな様子に、皆が気づいた。
Everyone noticed his manner that seemed to have some reason behind it.
故あり is the noun form meaning 'there is a reason'; 故ありげ adds the げ suffix to mean 'seeming to have a reason'.
Compound of 故 (yue, 'reason') + あり (ari, 'to exist') + げ (ge, suffix meaning 'seeming'). The exact historical derivation is uncertain, but the word follows a common pattern for archaic adjectival nouns.