This rare kanji has no clear standalone meaning. Treat its meaning as uncertain.
奒 is a rare character whose historical development is uncertain. It combines 大 (big) on top with 亥 below, but the original semantic or phonetic role of these components is unclear.
Since the meaning is unclear, focus on the shape: a big 大 sitting on top of 亥, like a large lid over a mysterious box.
For ガイ, imagine a guy named Gai lifting the big 大 lid off the 亥 box: Gai -> ガイ.