Understanding is often used in a countable sense. If you and I read the same book, we might understand it differently. So our understandings are not the same:
I have an understanding of the book, and you have an understanding of the book.
In your passage, the author imagines several different levels of understanding. They might look like this:
a basic understanding
an intermediate understanding
an advanced understanding
So the use of articles is natural in this context.
In your second sentence, a introduces sense, not understanding. But the article works similarly to the previous article. The author imagines that there can be a wide sense as well as a narrow sense and perhaps several kinds of sense between these two.
So this article is also natural.