Re: David Corfield's comment, this answer will mostly address "higher as in $(\infty, 1)$-categories" rather than "higher as in $n$-categories."

I also do understand you need the notion of abelian categories for homological algebra and constructing sheaves like the grand project done by Grothendieck.

This homological story is all about sheaves of abelian groups, or maybe sheaves of chain complexes; these are "linear" phenomena. As explained in David Ben-Zvi's excellent answer to the closely related MO question linked to in the comments, you can think of higher categories / homotopical algebra as a "nonlinear" generalization of homological algebra, suited to answering questions like the following:

What is a sheaf of categories on a space? Or a sheaf of homotopy types? Or a sheaf of cohomology theories?

There are many motivating examples here in algebraic geometry and algebraic topology. Here is a basic one: let $X$ be a topological space, let $G$ be a discrete group, and consider the presheaf sending an open set $U$ to the set of isomorphism classes of principal $G$-bundles on $U$. This presheaf is *not* a sheaf: for example, if $X$ is a manifold, then sufficiently small open sets are contractible and contractible spaces admit a unique isomorphism class of principal $G$-bundle, but the classification of principal $G$-bundles on $X$ nevertheless detects the fundamental group of $X$.

The problem is that in order to construct principal $G$-bundles on a space from principal $G$-bundles on an open cover of that space, we need to keep track of isomorphisms themselves and not just the relation of being isomorphic. This is a basic, fundamental conceptual move in higher category theory. That is, rather than a presheaf of sets we should consider a presheaf of *groupoids*, or homotopy $1$-types, sending an open set $U$ to the *groupoid* of principal $G$-bundles on $U$.

This presheaf *is* a sheaf, in a suitable sense (it is a stack, but I think that term scares people so I'll stick to "sheaf" and not even say "$2$-sheaf" or anything). (I need some niceness condition on $X$ here; finite CW-complex should suffice.) Because groupoids live in a $2$-category rather than a category, the sheaf axiom needs to be modified, and in particular it is now necessary to talk about triple rather than just double intersections. The correct sheaf condition is now that if $U = \bigcup_i U_i$ is an open covering, then a suitable diagram of the form

$$F(U) \to \prod_i F(U_i) \Rightarrow \prod_{i, j} F(U_i \cap U_j) \Rrightarrow \prod_{i, j, k} F(U_i \cap U_j \cap U_k)$$

exhibits $F(U)$ as the *2-limit* or *homotopy limit* of the rest of the diagram. Working out what this says about computing global sections from a nice open cover gets you precisely the Cech cocycle description of principal $G$-bundles.

(It's natural to want to allow $G$ to be a topological group here. Here we should really be keeping track of the entire space $[U, BG]$ of principal $G$-bundles on an open set $U$, and then the sheaf condition involves all finite intersections. But as a byproduct, by taking $BG$ to be an Eilenberg-MacLane space we recover the Cech cocycle description of higher cohomology.)

For an example closer to a modern direction of research, Lurie's approach to defining topological modular forms requires understanding what "the sheaf of elliptic cohomology theories over the moduli space of elliptic curves" means in a suitably derived sense, and then taking the global sections of this sheaf, again in a suitably derived sense.

For an example with relatively concrete geometric significance, one way to understand strong forms of the h-principle is that they answer questions of the following form: if $F, G$ are two presheaves of topological spaces on a manifold $X$ and $f : F \to G$ is a map of presheaves such that $F(U) \to G(U)$ is a (weak) homotopy equivalence for contractible open sets $U$, when can we conclude that $F(X) \to G(X)$ is a (weak) homotopy equivalence? (In particular, it does *not* suffice to assume that $F$ and $G$ are sheaves in the usual sense.) Motivating examples include the case that $F$ is the sheaf of immersions of $U$ into another manifold $Y$ and $G$ is the sheaf of "formal immersions": this leads to versions of the Hirsch-Smale theorem, as explained in these notes from a class by John Francis, that tell us some very concrete things about when manifolds can be immersed into, for example, $\mathbb{R}^n$.

(To be clear, I'm not claiming that you need to understand higher categories in order to understand the Hirsch-Smale theorem; certainly Hirsch and Smale did not need them. I just wanted to give an example where 1) people look at and care about presheaves of spaces and maps between them and 2) the usual sheaf condition doesn't answer the local-to-global question you want to answer.)

I have talked to several top mathematicians in my department, many of whom apperently think it is useless.

It's worth pointing out that mathematical culture differs greatly from department to department.

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