Literally, it means "not to leave X without something happening to X"
By using with words expressing emotion, the person naturally or necessarily causes something to happen..
The adverbs かならず necessarily, certainly 、 まちがいなく without fail、 ぜったいに certainly, surely, absolutely can be used at the place ofずにはおかない and have the same meaning

It is often used to refer to spontaneous actions or effects, such as a change in emotion or the outbreak of conflict. "Doing (something) definitely" Expresses the speaker's strong conviction to do so or to never something end without doing so. It may also be used to express strong emotion for acts or situations the speaker deems unacceptable, unforgivable, or inexcusable if they continue without without something being done about it. I'm wondering if in certain negative contexts it may be similar to English expressions like, "This (activity/ state) won't cannot stand/ continue/ keep going on." Possible translations (depending on the context; be careful here) also include "compelling" (人を感動させずにはおかない小説 a compelling novel) and "unfailing," among others.

おかない here seems to come from the word 措く、which means "to leave as is" or "to leave alone."

So basically, Vずには措かない basically means "(Subject) won't leave things alone as long as V isn't done." AKA, "V must be done."

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Examples

• あの やろう、次会ったら説教せずにはおかないぞ。
  • That bastard. Next time I see him, I won't leave things alone without teaching him a lesson.
  • That bastard. Next time I see him, I will definitely teach him a lesson.
• 火山灰はその村に被害を与えずにはおかないだろう。
  • The volcanic ash will not leave that village unaffected.
  • The volcanic ash will definitely do damage to the village.
• 彼は何事もせずにおかない。
  • He won't leave things alone without doing everything.
  • He will definitely do everything.
  • He never leaves anything undone.

必ず自分がAする、必ず相手にAさせるという強い決意を表す