noun
moment when blossoms fall; scattering of petals
Literally the instant when flowers, especially cherry blossoms, scatter. Often used in poetic or reflective contexts about transience.
桜の散り際は、はかなくて美しい。
The moment when cherry blossoms fall is fleeting and beautiful.
noun
one's last moments; on the point of death; deathbed
Figurative extension from falling blossoms: the way one faces death, or the final moments of life. Often carries a nuance of dignity or grace in one's final moments.
彼の散り際は、まるで武士のようだった。
His final moments were just like those of a samurai.
Compound of 散り (chiri, continuative form of 散る 'to scatter, fall') + 際 (giwa, 'time, verge'). The figurative sense of 'one's last moments' comes from the image of falling cherry blossoms.