Use it to add a more significant or surprising point after listing other things. He's talented at music and art, to say nothing of his exceptional writing skills.
Pro Tip 2/3
Emphasis Effect
Creates an effect by emphasizing that the additional information is even more noteworthy. She speaks four languages, to say nothing of her fluency in Mandarin, which is especially difficult.
Pro Tip 3/3
Implicit Importance
The phrase hints that the additional information gives more weight or value to the sentence. The company's profits rose, to say nothing of the customer satisfaction which skyrocketed.