How to think of the sheafification Let $\mathcal{F}$ be a presheaf of abelian groups over a topological space $X$, and one can construct the shefification of $\mathcal{F}$ by defining it to be the sections of $X \to \tilde{\mathcal{F}}$, where $\tilde{\mathcal{F}}$ is the sheaf space associated with $\mathcal{F}$. The definition is perfectly clear, but is really hard to work with. For instance one can look at the sheaf morphism $d:C^{\infty}(M) \to \Omega^{1}(M): f \to df$. We know that the image is not necessarily a sheaf, but I am having a really hard time trying to get an intuition about what the shefification of its image should be.
 A: There are two things we require of a presheaf in order to call it a sheaf:

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*Gluing: If $U$ is an open set with an open cover $\{U_i\}_{i\in I}$ and sections $s_i\in \mathcal{F}(U_i)$ so that $s_i|_{U_i\cap U_j} = s_j|_{U_i\cap U_j}$, then there's a section $s\in \mathcal{F}(U)$ so that $s|_{U_i}=s_i$.

*Locality: If we have two sections $s,t\in\mathcal{F}(U)$ and an open cover $\{U_i\}_{i\in I}$ so that $s|_{U_i}=t|_{U_i}$ for all $i$, then $s=t$.

Sheafification is the process of turning a presheaf into a sheaf by enforcing these relations in the most natural way we can. This process has two main nice features: it preserves the stalks, and any map from a presheaf to a sheaf factors through the sheafification.
If you're specifically interested in the case of the exterior derivative on a manifold, here's one example of something that's in the sheaf image but not the presheaf image. Consider $S^1$ and the differential form $d\theta$: this is not globally $d$ of anything, but locally it is: on any simply-connected set, we can find a function $f$ which has $df=d\theta$ on that set by integrating. These sections don't glue in the presheaf, but they do in the sheaf.
If you're looking to get a handle on using sheafification, the most natural ways to approach it are as an adjoint functor to the inclusion of presheaves in to sheaves, and the fact that it preserves stalks. In particular, one problem solving strategy you can use is if you're defining a map from the sheafification of some presheaf and you want to verify it has some property, checking on stalks can remove some hurdles because stalks don't change on sheafification.
Let me also make a (hopefully reassuring) comment that typically one does not have to worry about sheafification of this kind much in their life. There are a few weeks when one learns what it is and proves a few basic results where it's important, and then it only comes up sporadically.
