。
She suddenly couldn't help bursting into tears.
Literal
She [topic-は] suddenly burst-out-crying-not [without-ずには] could-not-be.
~ずにはいられない is one of Japanese's classic 'cannot help but' constructions, formed from the negative continuative ~ず + には + the negative potential of いる. The literal logic: 'cannot exist without [doing X],' i.e. cannot keep oneself from X. It's stronger and more written than the colloquial ~ずにはおれない and the equally formal ~ないではいられない, but all three say the same thing. Stacking ~出す inside the construction (泣きだす rather than 泣く) makes the verb specifically 'burst out crying,' so the whole expression reads as 'couldn't keep from bursting into tears.' That double layer of inevitability is what gives the sentence its emotional weight.