Learn German: Er kann die Geige nicht halten, geschweige denn fehlerfrei spielen. - He cannot hold the violin, let alone play flawlessly.

It functions like a coordinating conjunction (such as und or oder) connecting two parallel phrases. Because it connects the infinitives halten and spielen, it doesn't change the verb position. The subject and modal verb (Er kann) apply to both parts, so they don't need to be repeated.

Why is geschweige denn preceded by a negative word (nicht) in the first half of the sentence?

geschweige denn (let alone) is strictly used to follow up a negative statement. You use it to emphasize that because a basic or easier action (nicht halten) is impossible, a more advanced action (fehlerfrei spielen) is completely out of the question.

Why is the verb spielen placed at the very end?

The modal verb kann pushes the main verbs to the end of their respective clauses. If we wrote out the full second clause, it would be geschweige denn kann er sie fehlerfrei spielen. Since we drop the repeated words, we are left with the infinitive spielen right at the end.

Is fehlerfrei an adverb or an adjective here?

It functions as an adverb here because it describes how he plays the violin (flawlessly). In German, most adjectives and adverbs share the exact same form. It's built from der Fehler (mistake) and the suffix -frei (free from), literally meaning mistake-free.