The -form of a verb V is usually translated as "without doing V". In that sense, it has a similar meaning to V.
It is formed by taking the -form stem, but instead of adding , you add .
So, here: (dict.f.) → (nai-f.) → (nai-f. stem) → (zu-f.).
From what I understand, it's a hangover of a negative form used more commonly in classical Japanese. In modern Japanese, it is mainly used in writing, rather than speech, where V would be preferred.
Hope that helps!