What it means
ずにはいられない expresses an unavoidable emotional reaction or urge to perform an action. It is used when the speaker feels that they cannot keep from doing something or cannot prevent a feeling from arising. It attaches to the negative stem of a verb, and the polite counterpart is ずにはいられません. The construction often describes a strong personal response, although it can also describe another person's reaction when the context makes that clear.
Usage note
This form attaches to verbs, not directly to nouns or adjectives. For する-verbs, the form is せずにはいられない, as in 感動せずにはいられない. The ending can be made polite with ずにはいられません, without changing the preceding verb form.
How to use it
新しい映画を見ると、感動せずにはいられません。
When I watch a new movie, I cannot help but be moved.
新しい映画を見ると、感動せずにはいられない。
When I watch a new movie, I cannot help but be moved.