What it means
ないではいられない expresses that someone cannot refrain from doing something because of a strong emotional, physical, or psychological impulse. It is commonly used when a feeling naturally leads to an unavoidable reaction, such as crying, laughing, worrying, or thinking about something. The verb before it must be in the negative nai-form, but the overall meaning is positive: the person cannot remain without doing that action.
Usage note
ないではいられない focuses on an involuntary urge or reaction. Do not confuse it with ないわけにはいかない, which usually expresses an external obligation or a situation in which someone cannot avoid doing something because of duty or circumstances. ずにはいられない is a closely related alternative, although ないではいられない is often more direct and conversational.
How to use it
私は映画を見ると、涙を流さないではいられません。
Whenever I watch a movie, I cannot help but shed tears.
私は映画を見ると、涙を流さないではいられない。
Whenever I watch a movie, I cannot help but shed tears.