If a double had happened to give 14 significant figures the article would list the exact same "why" 14 is enough. I.e. the article explains why it's more than enough, it doesn't explain why it's the exact number of digits used.
If they had said X digits you'd be asking "but why X digits?", for any X.
Precisely why I said it'd be good if the article had answered that question. The value of X isn't happenstance in this case, the article just doesn't mention that.