It’s trying to give an intuition and nothing more so, if the passage is not helpful to you, just skip it. Look at the actual definition instead.
Honestly, I think it’s a very bad and unhelpful attempt at giving intuition. $\lim_{n\to\infty}f(n)=g(n)$ says “I don’t care how it gets there but, for large enough $n$, $f$ gets arbitrarily close to $g$,” whereas $f(n)=O(g(n))$ says, “I don’t care how it gets there but there’s a constant $c$ such that, for large enough $n$, $f$ is no bigger than $cg$.” So, really, big-O suppresses more than limit and the quote is wrong.
And don’t worry about the “suppressing a number” stuff, either. Saying $3x=O(x)$ suppresses the number three but $3x+1=O(x)$ suppresses two numbers, $3x=O(3x)$ suppresses nothing and $x=O(3x+1)$, er, un-suppresses stuff. So that’s wrong, too.