The second half of the article answers the question of how CBT addresses sleep problems.

Where? It describes some effects of CBT may have but nowhere does it describe what CBT treatment actually entails.

It links both to an academic paper and an online self-help thing prescribed by his doctor, and it talks at a high level about what CBT is. Are you familiar with CBT? If you aren't, the trouble you might be having is that CBT is pretty subtle: it's (in loose terms) a set of exercises you perform to train your mind to respond differently to stimuli. If you've ever read David Foster Wallace's Kendall College Commencement, when he talks about the "default setting", he's giving a pretty decent layman's explanation of the goals of CBT.

My point is that when reading an article in a publication like the New York Times, one should not need to follow links to academic papers or be familiar, a priori, with the subject matter. Providing a high-level overview without requiring the reading of additional sources is precisely what a properly written article in a newspaper like this is supposed do. This article fails at that quite badly.