break, smash, broken, busted
碎 centers on forceful destruction into pieces: breaking, smashing, or being broken and busted. The meanings all describe something shattered or reduced to fragments.
碎 combines 石 (stone) with 卒, which likely contributes the sound and the idea of finishing or ending, suggesting something broken into pieces by a stone.
A stone 石 on the left and 卒 on the right: imagine a stone hammer smashing something into fragments, finishing it off.
For サイ, picture a sigh of frustration as a stone breaks something: sigh -> サイ, and the stone smashes it to bits.