"Is it describing a subset of natural numbers, excluding 1, that is the product of two other natural numbers, of which one must be 1? Isn't that just every natural number except 1?"
Almost. It is saying that if $n = ab$, no matter which $a$ or $b$ you choose, it must be that either $a$ or $b$ is $1$.
Example: $6$. Is $6$ in the set? Well $6 \ne 1$ so that's a start. And "if $6 = a*b$..." Okay, that could be $(a,b) = (1,6),(2,3),(3,2)$ or $(61)$ "it must be that $a = 1$ or $b = 1$". Well... we could have $a = 2$ and $b=3 $ and neither $a$ nor $b$ must be $1$. So $6$ is not in the set.
SPOILER ALERT: There is a word for numbers that must be in the set--- that is numbers for whom every pair of divisors must include $1$... numbers that can not be a product of any pair of numbers of which neither are $1$....