cover, lid, flap
盖 centers on something that closes or shields an opening: a cover, lid, or flap. The idea is a flat or hinged piece that conceals or protects what is underneath.
盖 is a character of Chinese origin, combining 皿 (dish, vessel) with a phonetic element above. The upper part is related to 羊 (sheep) and 并, but the exact historical development is uncertain. The character conveys the idea of a cover or lid, likely from the image of something placed over a vessel.
The bottom 皿 is a dish, and the top looks like a flat lid with a handle. Imagine placing a cover over a dish to keep its contents hidden: 盖 is a cover or lid.
For ガイ, picture a guy placing a lid on a pot: guy -> ガイ, and he covers the dish with a flap.