In Proof that sheafification induces isomorphism on stalks using adjoints, Zhen Lin affirms that we can use adjointness to prove that the sheafification preserves stalks in the following way.
Concatenating the sheafification / inclusion (on the right) and skyscraper / stalk (on the left) adjuntions
we get that the functor $\sf{C}_X^\text{pre}\to\sf{C}$ which takes a presheaf, sheafifies and then takes the stalk is left adjoint to the skyscraper sheaf functor $\sf{C}\to\sf{C}_X^\text{pre}$. By unicity, it suffices to show that stalk functor $\sf{C}_X^\text{pre}\to\sf{C}$ is also a left adjoint of the skyscraper functor.
I would like to know why this adjunction holds.
Little discussion: let's recall how one could prove that the adjunction on the left of my diagram holds. Taking the adjunction between direct and inverse images in the case of the inclusion $i_p:\{p\}\to X$ of a point in $X$, we get
Now, the category $\sf{C}_{\{p\}}$ is equivalent to $\sf{C}$ since defining a sheaf on $\{p\}$ is the same thing as giving an object of $\sf{C}$ (as the image of $\varnothing$ is automatically defined). This proves the adjunction on the left above. Nevertheless, this same proof doesn't seems to work when working with presheaves since $\sf{C}_X^\text{pre}$ is not equivalent to $\sf{C}$. (The image of $\varnothing$ is not necessarily a final object of $\sf{C}$ in this case.)
As requested, the two comments above are turned into an answer.
You need to prove the adjunction directly. Note that a morphism of presheaves $A\to {i_p}_*B$ is the data for all U of a morphisms $A(U)\to{i_p}_*B(U)$ such that some diagrams commute. For all $U\not\ni p$, there is no data here since ${i_p}_*B(U)$ is the terminal object of your category and thus there is a single possibility for $A(U)\to{i_p}_*B(U)$. It follows that the morphism $A\to {i_p}_*B$ is equivalent to the data for all $U\ni p$ of $A(U)\to {i_p}_*B(U)$ such that some diagrams commute which in turn is equivalent to a morphism $A_p\to B$ by definition of the colimit.
This is basically the same proof that the stalk functor and the skyscraper functor are adjoints in the category of sheaves, the proof does not uses the sheaf condition. In fact, the proof for presheaves implies the proof for sheaves : $$Hom_{Sh}(A,{i_p}_*B)=Hom_{PSh}(A,{i_p}_*B)=Hom(A_p,B)$$ And you have that the sheaffification commutes with stalks : $$Hom(A^a_p,B)=Hom_{Sh}(A^a,{i_p}_*B)=Hom_{PSh}(A,{i_p}_*B)=Hom(A_p,B)$$ which implies that $A^a_p=A_p$ by Yoneda.