@MauroAllegranza gets at why we need second-order quantification, let me say a bit about why some want this to be interpreted as introducing plural quantification.
The main motivation for proponents of plural logic — why they want plural quantification instead of singular quantification over sets or properties — is ontological/conceptual. Their thought is that representing, e.g., the cereal on your spoon as a set distorts what’s intuitively the case: you’re eating bits of cereal, not a set of cereal.
There’s also a dialectical component. Insofar as second-order quantification involves quantification over sets or properties it has ontological commitments — sets or properties have to exist. Some, like Quine, have thought that logic must be “ontologically neutral” and dismissed second-order logic as “set theory in sheep’s clothing”. Part of Boolos’s aim in introducing the plural interpretation was to get rid of these extra commitments. “Pluralities” are not (meant to be) additional objects over and above their constituents, whereas sets are.
Finally, some have hoped that a plural account of quantification could be use to avoid set-theoretic paradoxes that result from quantification over all sets if the domain of quantification is itself a set. On this question, see the subsection on Set Theory at the SEP article on Plural Quantification. Boolos originally mentioned this idea in passing in his "To be is to be the value of a variable (or some values of some variables)". The main proponent of this view is Timothy Williamson, who staked out the position in his 2003 article "Everything". (He has expanded on the view in subsequent works as well, such as his book Modal Logic as Metaphysics.)
Øystein Linnebo was an early critic of Boolos's suggestion, notably his "Plural Quantification Exposed" (2003). The objections can be framed in a variety of ways, Linnebo attacks the supposed ontological innocence of plural quantification. The crux of his criticism is the question of the coherence of higher-order plural quantification -- plural quantification over pluralities. He argues that a "full semantics" for plural quantification, like full second-order semantics more generally, needs a determinate notion of "arbitrary sub-plurality" (like arbitrary subset). It's well known that, in first-order ZFC, the powerset operation is not absolute and admits of varied interpretations (hence why you can establish the independence of the Continuum Hypothesis). If the Axiom of Constructibility is true, then V=L and "arbitrary subset" coincides with the constructible sets. But it's also consistent with ZFC that there are many more sets than the constructible sets -- in fact, that V != L seems to be the more prevalent view among those who think there's a unique universe of sets. Linnebo argues that the plural logician has given no reason to think "sub-plurality" is any more absolute/determinate. Essentially, the plural comprehension axiom is no more trivial than set/class comprehension and cannot be assumed to be ontologically innocent.
More generally, it might be worried whether the plurality/class distinction is one without a difference. Simply saying that a plurality isn't one thing isn't very reassuring considering that we can and do use singular terms and descriptions to refer to them -- cf. "the universe of sets". Even if you were to grant that maybe class-talk was best understood as plural-talk all along, there remains the fact that the orthodox ZFC-ist denies the existence of proper classes. They are a mere façon de parler, a view articulated by Allen Hazen in his 2012 "Reflections on Counterpart Theory" where he says that the proper ZFC-ist attitude towards proper classes is a sort of fictionalism where "esse est definiri" (to be is to be defined; a play on Berkeley's Idealist principle "esse est percipi","to be is to be perceived"). Such a view would place sharp limitations on the power of plural comprehension principle, essentially restricting you to a predicative plural logic. That predicative restrictions are needed is more or less explicitly conceded by Williamson. This just amplifies the need for a clear comprehension principle and clarification of the expressive power of plural logic -- the intuitive picture won't suffice.
In short: it gets complicated quickly and this is still an area of active research and controversy. I think most people who don't reject plural quantification as unintelligible -- which isn't an unpopular position, particularly among the older generation of philosophers who grew up with set-theory as the gold standard for regimentation -- find some intuitive appeal to the idea that domains of quantification could just be pluralities of the values of the variables. But while the literature on the topic presses the need for a precise comprehension principle, it seems that's where opinions diverge substantially -- to the extent that there are firmly held opinions. For some (disclaimer: I'm one of these folks), the lesson to take from various "paradoxes" like those generated by the Löwenheim-Skolem theorems (among others) is that our efforts at formalization are doomed to failure -- Haim Gaifman has a good paper on the significance of non-standard models, which explores these issues. While we can speak of "everything", any attempt to provide a well-behaved regimentation of this talk (particularly any recursive axiomatization) will fall short in one way or another.
