This is a really badly written passage. As written, it suggests the following statement (or something along these lines):
If $X$ is a set of cardinals, then $\vert X\vert$ is greater than every element of $X$.
This is wildly false. E.g. the set $\{\aleph_{n+1}: n\in\mathbb{N}\}$ is countable, but all of its elements are uncountable!
Instead, we have to use something specific about this particular "set" of cardinals. This comes down to the axiom (scheme) of replacement, or more specifically transfinite recursion (which is a consequence of, and indeed equivalent to, replacement). Given any ordinal $\alpha$, we can "iterate" the cardinal successor operation along $\alpha$ (taking unions at limit stages), starting with (say) $\aleph_0$; e.g. if we take $\alpha=\omega+1$, we get the sequence $$\aleph_0, \aleph_1, \aleph_2, ..., \aleph_\omega.$$ Now suppose $\kappa$ is an arbitrary cardinal. Then ZFC proves the existence of $\kappa^+$, and by iterating the cardinal successor operation along $\kappa^+$ (viewed as an ordinal) we get a set $S_\kappa$ of distinct cardinals of cardinality $\kappa^+$; but $S_\kappa\subseteq Card$, so $Card$ has cardinality $\ge\kappa^+>\kappa$, and we're done.
It may be worth at this point thinking about what set theory would look like without replacement. If you drop replacement from ZFC (resp. ZF), you get Zermelo set theory with choice, ZC (resp. without choice, Z). This theory is extremely weak; for example, it can't even prove the existence of the ordinal $\omega+\omega$ or of the set $V_\omega$!
In fact, it gets worse. The set $V_{\omega+\omega}$ is a model of ZC (this isn't hard to check), and the only cardinals it contains are $0, 1, 2, 3, ...,\omega$. This means that in fact $V_{\omega+\omega}$ believes that there is a set of all cardinals! This tells us:
The statement "the class of all cardinals is not a set" requires the axiom (scheme) of replacement.
...OK, this is misleading since what's really going on is that the usual definition of "cardinal" is the wrong one for Zermelo set theory. However, there is still a sense in which Replacement is necessary for the class of cardinals to be "as big as it should be": in some models of ZC, such as $V_{\omega+\omega}^L$, there are only "set-many cardinals" in the sense that there is some definable surjection from some set to the class of cardinals.
My point here really is that there is more to the statement "the class of cardinals is not a set" than may immediately be clear from the quoted passage.
