I had this thought when studying "Sheaf Theory through Examples" by Daniel Rosiak. In the introduction, there is the exploration of how one can fit together data of different sensors observing the same space. Here is a picture from page-11,
This system of mutually compatible local data assignments or “measurements” of the happenings on the space—where the various data assignments are, piece by piece, constrained by one another, and thereby patched together to supply an assignment over the entire space covered by the individual regions—is, in essence, what constitutes our sheaf.
Hmm this feels like the motif we have in Differential Geometry where we chart Manifolds. And now consider this section about the motivation behind the name of Sheaf:
Here one thinks of various regions as the parcels of an overall space covered by those pieces, the collection of which then serves as a site where certain happenings are held to take place, and the abstract sensors capturing local snapshots or measurements of all that is going on in each parcel are then regarded as being collected together into “stalks” of data, regarded as sitting over (or growing out of) the various parts of the ground space to which they are attached. A selection of a particular snapshot made from each of the intersecting regions) and collation (along unions of regions) of these sections captures how the various stalks of data are bound together.
pg-12,13
And this seems like the concept of a bundle in Differential Geometry!
What exactly is going on here? Is there a deep connection between Sheaf Theory and manifold Theory?
Sheaves are present in (real) differential geometry implicitly, but one does not usually need to develop the machinery until later (or not at all). Indeed, there are many types of smoothly varying tensor fields that one studies, such as differential forms. A smooth $k$-form on a smooth manifold $M$ over a neighborhood $U$ is a section of the bundle $\pi_U:\bigwedge^k (T^*M)|_U\to U$, where here a section is a geometric section; namely, a smooth map $s:U\to \bigwedge^k(T^*M)|_U$ such that $\pi \circ s =\mathrm{Id}_U$. We can define a sheaf $\Omega^k_X$ be $$U\mapsto \Omega^k_X(U)= \{\text{sections of}\:\pi_U\}.$$ It is not very hard to show that this is a sheaf. Note that for $k=0$, this is just the sheaf of smooth functions. The main feature of the smooth category is that one can escape defining sheaves essentially because all local objects can be globalized. Indeed, given a point $p\in U$ and a smooth function $f\in \mathcal{C}^\infty(U)$, there exists an open neighborhood $p\in V\subseteq U$ such that $f|_V$ extends to a global function on $M$. This extension lemma follows from the existence of smooth partitions of unity.
The trouble is when one begins working in the category of complex (or algebraic) manifolds. In this situation, a well known fact is that there are no global nonconstant holomorphic functions on a compact complex manifold. That is, if $\mathcal{O}_X$ is the sheaf of holomorphic functions on a compact manifold $X$, $\mathcal{O}_X(X) = \mathbb{C}$. By contrast, $\mathcal{C}^\infty(X)$ is infinite dimensional. One of the many benefits of working with sheaves is that they allow for definitions of many familiar objects from elementary differential geometry. For example, we can define an affine connection on a smooth vector bundle $\pi:E\to M$ to be an operator $$\nabla:\Gamma(M,\mathfrak{X}(M))\times \Gamma(M,E)\to \Gamma(M,E)\:\:\text{written}\:(X,s)\mapsto \nabla_Xs$$ that is $\mathbb{R}$-bilinear, and $\mathcal{C}^\infty(M)$-linear in the first variable, satisfying a Leibniz rule: $$\nabla_{X}(fs)=X(f)s+f\nabla_X s$$ where $f$ is a smooth function. One can define analogous notions of holomorphic (resp. algebraic) vector bundles on complex (resp. algebraic) manifolds, but the same definition will not work. Indeed, if $\mathcal{E}$ is a holomorphic vector bundle on a complex manifold $X$, and $\Theta(X)$ denotes the space of global holomorphic vector fields, there is no reason to believe that $\Gamma(X,\mathcal{E})\ne 0$ and similarly for $\Theta(X)$. Indeed, it is not hard to produce examples of Riemann surfaces $X$ with no global holomorphic vector fields. In that case, the connection would just be the zero map; not very interesting. A solution is to define a (holomorphic) connection on $\mathcal{E}$ to be a $\Bbb{C}$-linear morphism of sheaves: $$ \nabla: \mathcal{E}\to \mathcal{E}\otimes_{\mathcal{O}_X} \Omega^1_X $$ where $\Omega^1_X$ is the sheaf of holomorphic differential forms, satisfying a Leibniz rule: for all $U\subset X$, $s\in \Gamma(U,\mathcal{E})$, and $f\in \mathcal{O}_X(U)$ we have $\nabla(fs) = df\otimes s+ f\nabla s$. This is really a "dual" formalization of what it means to be a connection, but you can see the equivalence by evaluating on a holomorphic vector field $V$ over $U$. Anyway, this is one good reason to work with sheaves. You can define objects "open set by open set" which is conceptually analogous to defining manifolds chart by chart.
There are a lot of other reasons you might like to work with sheaves. For one thing, if $X$ is a nice topological space (think manifold or CW complex), then the category of covering spaces of $X$ is equivalent to the category of locally constant sheaves on $X$. One sends a cover $p:Y\to X$ to its associated sheaf; $$\mathcal{F}_Y(U) = \{s:U\to Y|p\circ s = \mathrm{Id}_U\}.$$ You might have noticed a similarity to the original example: a sheaf corresponds to a geometric object mapping to $X$ by taking replacing the geometric object with a sheaf of sections. Indeed, this is no coincidence.
Let $(X,\mathcal{O}_X)$ be a complex manifold with its sheaf of holomorphic functions. There is an equivalence between locally free sheaves of $\mathcal{O}_X$-modules and holomorphic vector bundles; a locally free sheaf of $\mathcal{O}_X$-modules is a sheaf $\mathcal{F}$ with an $\mathcal{O}_X$-module structure (there are multiplication maps $\mathcal{O}_X(U)\times \mathcal{F}(U)\to \mathcal{F}(U)$ etc.) such that there is an open cover $\{U_\alpha\}$ of $X$ where $\mathcal{F}|_{U_\alpha} \cong \mathcal{O}_X^{\oplus n}|_{U_\alpha}$ for some $n$. The idea is that if $\mathcal{E}|_{U_\alpha}\cong \mathbb{C}^n\times U_\alpha$, there are $n$ distinguished sections; $s_i:U_\alpha \to \mathcal{E}|_{U_\alpha}\cong \Bbb{C}^n\times U_\alpha$ by $s_i(p) = (e_i,p) = ((0,\ldots, 1,\ldots,0),p)$. These functions define the isomorphism locally.
There's a lot more to be said, but I'll cut it off here.