Use "not to mention ___" when you're listing reasons or excuses. It marks things that are kind of "extra" items on the list.
For example, in this sentence:
Smoking is a nasty habit – bad for your health, expensive, not to mention the odor.
...the speaker gave two good reasons why smoking is a bad habit: it's bad for your health and expensive. The bad odor that cigarette smoke leaves is another reason, but it's not as strong of a reason as the first two. So the speaker adds that to the list with "not to mention".
The costs, not to mention the risks, of transporting the telescope to and from space would have been excessive.
You can follow "not to mention" with a noun like in the examples above. You can also follow it with a sentence or clause:
It's so much easier to order something than to spend all that time going to the grocery store, cooking, washing dishes. Not to mention I'm not that good of a cook to begin with.