What does it mean for a sequence of sheaves to be exact

The image of a map is defined to be the sheafification of $Im^{pre}$. By the universal property of the sheafification this sheaf is unique up to unique isomorphism. Now as you already mentioned there is a natural injection $i: Im(\alpha) \rightarrow G$. We can now define the above sequence to be exact if $i(Im(\alpha))=Ker(\beta)$.

Edit: To see why this works assume the two sheafs $Im(\alpha)$ and $Im(\alpha)^{'}$ are both sheafifications of the sheaf $Im^{pre}(\alpha)$(with natural maps $\theta$ and $\theta ^{'}$). If $j:Im^{pre}(\alpha)\rightarrow G$ is the inclusion map and $\phi: Im(\alpha)) \rightarrow Im(\alpha))'$ the unique isomprhism, then $i \theta=j$. But since $i^{'} \theta ^{'}=j$ and $\theta ^{'}=\phi \theta$, by uniqueness we have $i{'} \phi=i$. Thus $i(Im(\alpha))=i'(Im(\alpha '))$.

Let's take an example, a piece of the Poincare sequence on a smooth manifold $M$

$$ \Omega^k_M \overset{d_k}{\to}\Omega^{k+1}_M \overset{d_{k+1}}{\to} \Omega^{k+2}_M$$

It is exact at the term $\Omega^{k+1}(M)$. This means two things:

First, $d_{k+1}\circ d_k=0$.

Second, for every open subset $U$ of $M$ and $\omega \in \Omega^{k+1}_M(U)$ a $k+1$-differential form with $d \omega = 0$, there exists a covering $V_i$ of $U$ and $\eta_i \in \Omega^k_M(V_i)$ so that $d_k(\eta_i)= \omega_{|V_i}$ ( any closed form is ${\it locally}$ exact).