。
She nearly burst into tears.
Literal
[topic-は] almost cry-out-start [about-to-ところだった].
泣き出す is a compound verb: 泣く ('cry') + 出す ('start, begin emerging'), giving 'burst into tears, start crying [suddenly].' The auxiliary ~出す always carries that 'sudden onset' connotation — distinct from ~始める which is more neutral 'begin to.' 笑い出す ('burst out laughing'), 降り出す ('start raining, suddenly'), 走り出す ('break into a run') all share the pattern. The whole sentence is a vivid emotional snapshot rather than a finished narrative — common in slice-of-life fiction.