"There is nothing."
The も after interrogative pronouns like in 何も (also in いつも, だれも, どこも, probably others) has a different function to the も used after nouns generally to mean "also". In this case with a negative verb 何も expresses the concept of "anything", so 「何もありません」 means literally "there is not anything" or equivalently "there is nothing". Similarly いつも expresses "anytime" with a negative verb, だれも expresses "anybody" with a negative verb, etc. Some of the interrogatives+も (but not 何も for some reason) can also be used with positive verbs and in those cases expresses the complete category e.g. "always", "everybody".
I know すべて is one word for everything. But context matters I suppose.
I'll leave everything to you
On its own, ありません just means "there isn't". That's not the same as "there isn't anything" (there is nothing). For example, if someone asks you 「カバンにペンがありますか？」(is there a pen in (your) bag?) you could answer 「ありません」(there isn't.) If you instead answered 「何もありません」 (there isn't anything) the meaning is very different, as though your bag is completely empty.