She aired her complaints and grievances, and began to cry.
Literal
She [topic-は] complaints [object-を] spilled, crying began.
不平不満 is a four-character compound doubling 不平 and 不満 for emphasis — 'complaints and grievances.' こぼす means 'to spill/let slip.' 泣き始めた uses ~始める for the onset of crying.