When I heard of my mother's death, I couldn't help but cry.
Literal
Mother [possessive-の] death [object-を] heard-[sequential-て] cry [without-ず] [topic-には] could-not-be
~ずにはいられない = 'cannot help but do X' — a grammar pattern expressing an irresistible impulse. ず is the classical negative, には marks the condition, いられない (potential negative of いる) = 'cannot remain.' The whole pattern means the emotion was too strong to suppress.