How should I think about presentable $\infty$-categories?

Presentable $\infty$-categories can be understood without every having to think about cardinals. An $\infty$-category is presentable iff it is equivalent to one of the form $\mathcal{P}(C,R)$, where

• $C$ is a small $\infty$-category,
• $R=\{f_i\colon X_i\to Y_i\}$ is a set of maps in $\mathrm{PSh}(C)=\mathrm{Fun}(C^\mathrm{op}, \mathrm{Gpd}_\infty)$, and
• $\mathcal{P}(C,R)$ is the full subcategory of $\mathrm{PSh}(C)$ spanned by $F$ such that $\mathrm{Map}(f,F)$ is an isomorphism of $\infty$-groupoids for all $f\in R$.

That's it. The conditions that $C$ is small and $R$ is a set allow you to show that the inclusion $\mathcal{P}(C,R)\to \mathrm{PSh}(C)$ admits a left adjoint, which implies that $\mathcal{P}(C,R)$ is complete and cocomplete, which is something you definitely want.

"Presentable" should be thought of in terms of "presentation", analogous to presentations of a group. In some sense $\mathcal{P}(C,R)$ is "freely generated under colimits by $C$, subject to relations $R$". More precisely, there is an equivalence between (i) colimit preserving functors $\mathcal{P}(C,R)\to D$ to cocomplete $\infty$-category $D$, and (i) a certain full subcategory of all functors $F\colon C\to D$ that "send relations to isomorphisms" (precisely: those $F$ such that $\widehat{F}(f)$ is iso for all $f\in R$, where $\widehat{F}\colon \mathrm{PSh}(C)\to D$ is the left Kan extension of $F$ along $C\to \mathrm{Psh}(C)$).

So its easy to construct colimit preserving functors from presentable categories (and all such functors turn out to be left adjoints).

Here is a naive answer. Set theoretical issues play a role, sure, but that's not how I think of presentable $\infty$-categories (or presentable categories!). Really, the key feature of presentable ($\infty$-)categories is that their objects are "presentable". You have a set of generators, and every object of you category can be written as a "small" colimit – with a nice shape – of these generators. Because colimits are what they are, you can easily compute morphisms out of such a colimit. And because the generators are "compact", you can compute morphisms out of a generator into the nicely-shaped colimits easily too. So in the end, all you need to care about are your generators: any object is a nice colimit of the generators, and any morphism can be expressed in terms of morphisms between the generators.

Maybe examples are best. The category of sets is presentable. Any set is a nice colimit of its finite subsets, and any map from a finite set into a nice colimit factors through one of the factors of the colimit. So to study sets, you can just study finite sets, and only think about finite subsets of bigger sets (that's what we do all the time!). Another example would be modules: any module is a nice colimit of finite-rank free modules. You can do a lot of stuff with finite-rank free modules, and only then think about bigger modules and quotients.

Of course, size issues play a big role; if you don't impose them, then nothing would prevent you from choosing all objects as generators, which is somewhat useless...

The nLab article is nicely written if you want more details.

Presentable $\infty$-categories also have a very strong point in their favor from my limited perspective: they are exactly the one that come from combinatorial simplicial model categories (see e.g. Proposition A.3.7.6 in Lurie's HTT). For people interested in homotopy theory, that's pretty sweet. You can in some sense think that the $\infty$-category in question is "presented" by the model category: using all the machinery, you can explicitly compute hom spaces that perhaps you couldn't with just the $\infty$-category. And if you are convinced that model categories are interesting, and if you are also convinced that locally presentable categories are interesting (since combinatorial model categories are locally presentable by definition) then surely you would agree that this makes presentable $\infty$-category interesting, the previous statement being an if and only if.

PS: Some care about the terminology is needed. A presentable ($\infty-$)category is not a presentable object in the category of categories. Rather, the objects of the category are presentable, and many authors say "locally presentable category" instead (like "locally small category": the category isn't small, but if you focus on two objects at a time, it is).

Just to be a little bit contrary, let me point out one concrete reason that locally presentable categories are not aesthetic: it is unclear whether they have any analogue in constructive mathematics. Most of basic category theory is entirely constructive (the classic quip is "have you ever seen anyone prove that a diagram commutes by assuming that it doesn't and deriving a contradiction?"). But once you move into locally-presentable world, all that goes out the window: the well-ordering and transfinite iteration techniques, and the theory of ordinal and cardinal numbers that underlie them, rely relentlessly not only on the law of excluded middle but also the axiom of choice. It's not impossible that there is an analogous theory constructively, but as far as I know, no one has ever managed to write one down.