For me the sheafification of a given presheaf is this:
Proposition-Definition: Given a presheaf $\mathscr{F}$, there is a sheaf $\mathscr{F}^+$ and a morphism $\theta \colon \mathscr{F} \to \mathscr{F}^+$, with the property that for any sheaf $\mathscr{G}$, and any morphism $\varphi \colon \mathscr{F} \to \mathscr{G}$, there is a unique morphism $\psi \colon \mathscr{F}^+ \to \mathscr{G}$ such that $\varphi = \psi \circ \theta$. Furthermore the pair $(\mathscr{F}^+, \theta)$ is unique up to unique isomorphism. $\mathscr{F}^+$ is called the sheaf associated to the presheaf $\mathscr{F}$.
I know that this is equivalent to a universally repelling object in a certain category. But how could this be equal to the following?
Suppose $\mathcal{F} \subset \mathscr{S}$, where $\mathscr{S}$ is a sheaf. Then we define $$ \mathcal{F}^+ = \left\{ s \in \mathscr{S}(U) \,\middle|\, \forall U \in \tau (X) : \text{$s$ is locally in $\mathcal{F}(U)$} \right\} $$ for a fixed topological space $X$. By locally in $\mathcal{F}(U)$ we mean that given $s \in \mathscr{S}(U)$ there exists an open covering $\{U_{\alpha}\}_{\alpha \in I}$ of $U$ such that $s|_{U_\alpha} \in \mathcal{F}(U_{\alpha}) \subset \mathscr{S}(U_{\alpha})$.
The thing is that the only tool I have so far is the definition, so how can I get this? I know that I have to verify the universal property of my new $\mathcal{F}^{+}$ but I don't know how to do this.
Thanks a lot in advance
Your highlighted definition intuitively says that the sheafification $\mathcal{F} \rightarrow \mathcal{F^{+}}$ only does what every map from $\mathcal{F}$ to a sheaf must do. Literally: sheafification adds just enough new sections to $\mathcal{F}$ to make a sheaf, and merges together just enough sections of $\mathcal{F}$ -- so every map from $\mathcal{F}$ to a sheaf factors through the sheafification.
The second definition uses a presumed embedding $\mathcal{F} \subset \mathscr{S}$ into a sheaf to say explicitly what sections must be added. If $\mathcal{F}$ has an embedding into a sheaf, then no sections need to be merged.
But of course the sheafification might not be all of $\mathscr{S}$. It will use just those sections in the set
$$\{s \in \mathscr{S}(U) \; \big{|} \; s \;\text{is locally in} \; \mathcal{F}(U)\; \forall U \in \tau (X) \}$$