Learn Dutch: Zij blijft niet rustig bij een wesp, laat staan dat ze haar paniek verbergt. - She does not stay calm near a wasp, let alone hide her panic.

Because laat staan dat (let alone that) acts as a subordinating conjunction. The word dat triggers subordinate clause word order, pushing the conjugated verb (verbergt) to the end of its clause.

Could I leave out dat and just use laat staan here?

You can only drop dat if you aren't using a full clause with a conjugated subject and verb. If the sentence was ..., laat staan haar paniek verbergen (using the infinitive) or just ..., laat staan haar paniek (a noun phrase), you would drop dat. But because this uses the conjugated verb verbergt with the subject ze, dat is required.

Does laat staan dat always need to follow a negative statement?

Yes! Just like 'let alone' in English, laat staan (dat) only makes sense if the first part of the sentence establishes a negative premise. You need a negative word like niet or geen in the main clause to logically say 'X does not even happen, let alone Y.'

Why use the preposition bij for 'near a wasp'?