The set of numbers that can be written with a finite number of 0-9 digits is countable. That set is called the decimal numbers, and is a subset of the rational numbers which itself is countable too.
If you allow infinitely long descriptions, then I don't know what's the score. In the case of "descriptions" that amount to (potentially infinite) decimal expansions, I think an argument akin to Cantor's diagonal will show that you can't possibly describe all real numbers (or all numbers between 0 and 1), but I'm not confident I can assert this to be true.
That doesn’t sound like the claim the author is making in the sentence I quoted but maybe I am being uncharitable?