dry in sun, expose to sun
曬 centers on exposing something to sunlight, especially to dry it out. The core idea is sun-drying or airing in the sun.
曬 combines 日 (sun) with 麗, which likely contributes the sound and may have originally depicted a deer under the sun, reinforcing the idea of exposure to sunlight. The exact historical development is uncertain.
The sun 日 on the left shines on the right side 麗, like sunlight pouring over something to dry it. Picture hanging laundry under a bright sun to dry.
For サイ, imagine drying clothes under a bright sun while sighing with relief: sigh -> サイ, as the sun dries everything.