The classical lore is that $H^1(X,\mathcal F)$ is obstruction to lifting local data to global data. However I don't understand why one would want to compute $H^3(X,\mathcal F), H^4(X,\mathcal F), \cdots$.
For complex manifold $X$, $H^1(X,\mathcal O),H^{0,1}(X)$ both represent obstruction to local-to-global lifting of holomorphic functions. This in particular allows one to determine whether Mittag-Leffler problem can be solved. $H^1(X,\mathcal O)=0$ implies local solutions can be modified to identify a global solution, and $H^{0,1}(X)=0$ implies that local solutions can be multiplied by a smooth bump function, after which $\bar\partial$-exactness kicks in to save the day.
However, what does $H^q(X,\Omega^p)$ or $H^{p,q}(X)$ mean (which are the same by Dolbeault theorem)? $H^1(X,\Omega^p)$ or $H^{p,1}(X)$ should represent the same local-to-global problem for holomorphic $p$-forms, but what about, say, $H^3(X,\Omega^p)$?
Sure, one can talk about Cech cohomology for a good cover; $H^3(X,\mathcal F)$, for example, is about lifting sections defined on quadruple-intersections to triple-intersections. That's fine and all, but that doesn't sound very compelling to me. It seems to me that lifting sections on $q$-fold intersections to sections on $(q-1)$-fold intersections doesn't really solve any natural, interesting problems that arise independently of the formalisms introduced (for example, Lefschetz fixed point formula solves the problem of counting fixed points, and this is defined independently of singular homology and therefore I'd consider this a very compelling reason to study singular homology groups).
Similar situation appears when we study Chern class and line bundles. The Bockstein morphism for exponential sheaf sequence $H^1(X,\mathcal O^\times) \rightarrow H^2(X,\mathbb Z)$ is precisely taking Chern class for line bundle, so it helps to know things like $H^1(X,\mathcal O)=H^2(X,\mathcal O)=0$, which allow us to classify line bundles for manifolds like $\mathbb {CP}^n$. However, there does not seem to be a reason to care about $H^3(X,\mathcal O)$, etc.
Note: I have checked out a similar question. Here, one of the answers point out that for sheaf $\mathcal F$, if we can find an acyclic sheaf $\mathcal A$ such that $\mathcal F$ is a subsheaf of $\mathcal A$, then
$$0\rightarrow \mathcal F \rightarrow \mathcal A \rightarrow \mathcal A/\mathcal F \rightarrow 0 $$
is exact and therefore by long exact sequence coming from this,
$$ H^{p}(X,\mathcal F) \cong H^{p-1}(X,\mathcal A / \mathcal F)$$
and therefore higher cohomology groups can be understood as obstruction ($H^1$), and actually even global section ($H^0$) of $\mathcal A_1/(\mathcal A_2 / \cdots (\mathcal A_p /\mathcal F)\cdots )$. This just mystifies the issue further for me, somewhat, largely because I can't think of a canonical choice of such acyclic $\mathcal A$ and therefore I can't interpret the meaning of local-to-global lifting of iterated-quotient sheaves.
Best Answer
I think these kinds of lifting-based statements are the wrong place to start. For me these arguments about lifting and such are most convincing as answers to questions like "What are cohomology groups?". For instance $H^1(X,\mathcal O_X)$ is the tangent space to the moduli space of line bundles on $X$. But that don't really explain why to study it. How much do you really care about the Mittag-Leffler problem? Probably not enough to base your entire mathematical career around it. But a huge fraction of modern mathematicians have based their mathematical careers around studying cohomology or generalisations of it.
Instead, as you suggest, what motivates the study of cohomology are its immensely powerful applications. For Dolbeaut cohomology and coherent sheaf cohomology in particular, I would say the premier applications are to classification-type problems in algebraic geometry.
First, Dolbeaut cohomology groups provide natural invariants (the Hodge numbers) that can be used to distinguish algebraic varieties.
Second, the machinery of sheaf cohomology is crucial in answering all sorts of geometric questions about line bundles and other geometric structures. A good example might be the proof of the Hodge index theorem via Hirzebruch-Riemann-Roch, as in Hartshorne. We need all the cohomology groups to define the Euler characteristic, and the Euler characteristic is such a nice invariant that there is a simple formula for it, and this simple formula helps us understand purely geometric questions about intersection of curves.
I would say that at a typical algebraic geometry seminar the majority of the talks involve higher sheaf cohomology at some point, but most talks are not devoted to answering a question expressed in terms of cohomology, so one could find many examples there.
Third, there are the applications via Hodge theory. The natural isomorphism between the singular cohomology and the Dolbeaut cohomology itself has nontrivial structure which is related in a highly nontrivial way to the geometry and arithmetic of the variety. In particular it is supposed to tell you about algebraic cycles - this is the Hodge conjecture. While that is open, many interesting facts are known - e.g. the K3 surfaces are completely determined by their Hodge structure.
Studying how these Hodge structures vary in a family of varieties leads to all sorts of interesting theory, which again can be used to solve purely geometric questions - the statement I remember off the top of my head is that varieties of general type cannot have a nowhere vanishing one-form (Popa and Schnell).
And this is not even getting into the very interesting applications of other cohomology theories for algebraic varieties, such as etale cohomology.