I begin by saying that while I understand what a triangulated / derived category is pretty well, I know nothing about Higher Algebra stuff and not even $\infty$-categories.
I've heard some people say that stable $\infty$-categories are a "more natural" point of view to derived categories. Can anyone explain in simple terms (given my background) to me why this is so?
I study algebraic and arithmetic geometry but all possible motivations are welcome.
I already answered some version of this question in this answer, but let me try to expand a bit on the concrete advantages in mathematical practice. For understanding the following you need to take on faith that ∞-categories exist and have roughly the same properties as ordinary categories (which in fact are just a special example of ∞-categories).
One of the biggest advantages of stable ∞-categories compared to triangulated categories is that they work well in families. This already should be appealing to an algebraic geometer. But it turns out that this makes constructing stable ∞-categories easier than triangulated categories (contrary to what you might be expecting). For example let me sketch a construction of the symmetric monoidal ∞-category of a scheme (you can easily adapt it to algebraic spaces/stacks etc..)
First you need to do the affine case: the derived category of a ring. There are many ways of doing it. The most concrete is probably taking the 1-category of chain complexes and inverting quasi-isomorphisms (in the ∞-categorical sense) $$\mathscr{D}(R):=\operatorname{Ch}(R)[q.iso^{-1}]$$ Then you need to prove that it is functorial, symmetric monoidal etc. I personally favour another (equivalent) definition, which uses more technology but it makes all properties follow formally. I am taking as the definition of the derived ∞-category the stabilization of the animation of the category of finitely generated projective $R$-modules $$\mathscr{D}(R):=\operatorname{Sp}(\mathcal{P}_\Sigma(\operatorname{Proj}_R))\,.$$ Then $\mathscr{D}(R)$ has all the properties we want because $\operatorname{Proj}_R$ does. Moreover you can still show easily that $\mathscr{D}(R)$ is localization of the category of chain complexes at quasi-isomorphisms by using a clever trick, so you lost nothing in concreteness.
Our next step is to show that $\mathscr{D}(-)$ as a functor on affine schemes satisfies Zariski descent. Concretely it boils down to showing that the following square of stable ∞-categories is cartesian $$\require{AMScd} \begin{CD} \mathscr{D}(R) @>>> \mathscr{D}(R[1/f])\\ @VVV @VVV\\ \mathscr{D}(R[1/g]) @>>> \mathscr{D}(R[1/fg])\end{CD}$$ and you can prove this is true again because it is true for $\operatorname{Proj}_R$ (where it is elementary).
NOTE THAT THIS IS FALSE FOR THE CORRESPONDING TRIANGULATED CATEGORIES
This is one of the amazing advantages of stable ∞-categories compared to triangulated categories: you can glue their objects. Using this now it is clear how to define the derived $\infty$-category of a scheme $X$ $$\mathscr{D}(X)=\lim_{\operatorname{Spec}R\subseteq X} \mathscr{D}(R)$$ where the limit is taken over the poset of open affine subsets of $X$. That is, we're saying that an object of $\mathscr{D}(X)$ is just the collection of an object of $\mathscr{D}(R)$ for each open affine subscheme plus suitable gluing data. Now by formal properties of limits this is automatically a symmetric monoidal $\infty$-category: we can take tensor products of elements in $\mathscr{D}(X)$ by taking them in any affine open and gluing back the resulting objects. If all you have are triangulated categories, constructing the symmetric monoidal structure on $h\mathscr{D}(X)$ is highly non-trivial.
One great advantage of stable $\infty$-categories is that stability is a property, not a structure. To construct a triangulated category it's not enough to construct the category: you also have to come up with a shift functor, and a family of exact triangles etc. In stable ∞-categories you don't have to worry about that: you just construct a certain ∞-category and then check that, say the (canonically defined) functor $\Omega$ is an equivalence. This has advantages because for example it's clear what should be a stable symmetric monoidal ∞-category: it's just a symmetric monoidal ∞-category which is stable and such that the tensor product is exact in each variable. Try instead to come up with the notion of symmetric monoidal triangulated category. I strongly suspect you would not come up with all the required axioms (there is a compatibility between the shift and the tensor product which has some tricky signs).
Triangulated categories rarely have colimits. They have direct sums and usually little more. Stable ∞-categories instead have all finite limits and colimits (and most of those you'll encounter in practice have all small limits and colimits). This gives you a huge flexibility in working in them. For example you can say that you can reconstruct the sheaves (say on a space $X$) from a sheaf on a neighborhood of a closed subset of $Z$ and a sheaf on the complement, plus some gluing data. This is classical for sheaves of abelian groups, but it gets tricky (although not impossible) to state for derived categories of sheaves. For derived ∞-categories, however, the same statement as for sheaves of abelian groups work, with the same proof.
A powerful motivation for me is that you cannot define the higher algebraic K-theory of a triangulated category. There are stable ∞-categories that have the same underlying triangulated category but not the same higher algebraic K-theory. As a person that finds algebraic K-theory very interesting, this would be enough for me to not want to work with triangulated categories altogether
You can actually have sheaves with values in an ∞-category. It makes perfect sense, say, to say that algebraic K-theory gives you a Zariski (or Nisnevich) sheaf of spectra on the category of schemes. This ended up allowing wonderful computations that would have been hard to do without stable ∞-categories. Perhaps not impossible, but most of the older work in the same vein used secretly something like dg-categories which has many of the same features (although not the same versatility), and in any case not triangulated categories.
