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.

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. In some ways, $\lim$ gives more information; in some ways, it gives less. In some senses, $\lim$ is more specific in that it gives a single value that the function asymptotically approaches, and the notion of limit makes sense approaching points other than $\pm\infty$. If the limit is a real number, then saying $\lim_{n\to\infty}f(n)=c$ gives more information than $f(n)=O(1)$, which suppresses $c$.

On the other hand, for many functions we're interested in, $\lim_{n\to\infty}f(n)=\infty$ so $\lim$ gives almost no information. In those cases, big-$O$ allows you to give more information.

I wouldn’t worry about the “suppressing a number” stuff. 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. Somethingl ike $x=O(x^2)$ is even less clear. It's hard to tell what de Bruijn really intended.