By Lemma \ref{lemma-gerbe-of-lifts-fibred} we see the first condition of Stacks, Definition \ref{stacks-definition-stack-in-groupoids} holds. As is customary we check descent of objects and we leave it to the reader to check descent of morphisms. Thus suppose we have $a : U \to W$ in $W_{spaces, \etale}$, a covering $\{U_k \to U\}_{k \in K}$ in $W_{spaces, \etale}$, objects $\xi_k = (U_k, U'_k, a_k, i_k, x'_k, \alpha_k)$ of $\mathcal{C}$ over $U_k$, and morphisms $$ \varphi_{kk'} = (f_{kk'}, f'_{kk'}, \gamma_{kk'}) : \xi_k|_{U_k \times_U U_{k'}} \to \xi_{k'}|_{U_k \times_U U_{k'}} $$ between restrictions satisfying the cocycle condition. In order to prove effectivity we may first refine the covering. Hence we may assume each $U_k$ is a scheme (even an affine scheme if you like). Let us write $$ \xi_k|_{U_k \times_U U_{k'}} = (U_k \times_U U_{k'}, U'_{kk'}, a_{kk'}, x'_{kk'}, \alpha_{kk'}) $$ Then we get an \'etale (by Lemma \ref{lemma-morphisms-lifts-etale}) morphism $s_{kk'} : U'_{kk'} \to U'_k$ as the second component of the morphism $\xi_k|_{U_k \times_U U_{k'}} \to \xi_k$ of $\mathcal{C}$. Similarly we obtain an \'etale morphism $t_{kk'} : U'_{kk'} \to U'_{k'}$ by looking at the second component of the composition $$ \xi_k|_{U_k \times_U U_{k'}} \xrightarrow{\varphi_{kk'}} \xi_{k'}|_{U_k \times_U U_{k'}} \to \xi_{k'} $$ We claim that $$ j : \coprod\nolimits_{(k, k') \in K \times K} U'_{kk'} \xrightarrow{(\coprod s_{kk'}, \coprod t_{kk'})} (\coprod\nolimits_{k \in K} U'_k) \times (\coprod\nolimits_{k \in K} U'_k) $$ is an \'etale equivalence relation. First, we have already seen that the components $s, t$ of the displayed morphism are \'etale. The base change of the morphism $j$ by $(\coprod U_k) \times (\coprod U_k) \to (\coprod U'_k) \times (\coprod U'_k)$ is a monomorphism because it is the map $$ \coprod\nolimits_{(k, k') \in K \times K} U_k \times_U U_{k'} \longrightarrow (\coprod\nolimits_{k \in K} U_k) \times (\coprod\nolimits_{k \in K} U_k) $$ Hence $j$ is a monomorphism by More on Morphisms, Lemma \ref{more-morphisms-lemma-properties-that-extend-over-thickenings}. Finally, symmetry of the relation $j$ comes from the fact that $\varphi_{kk'}^{-1}$ is the ``flip'' of $\varphi_{k'k}$ (see Stacks, Remarks \ref{stacks-remarks-definition-descent-datum}) and transitivity comes from the cocycle condition (details omitted). Thus the quotient of $\coprod U'_k$ by $j$ is an algebraic space $U'$ (Spaces, Theorem \ref{spaces-theorem-presentation}). Above we have already shown that there is a thickening $i : U \to U'$ as we saw that the restriction of $j$ on $\coprod U_k$ gives $(\coprod U_k) \times_U (\coprod U_k)$. Finally, if we temporarily view the $1$-morphisms $x'_k : U'_k \to \mathcal{X}'$ as objects of the stack $\mathcal{X}'$ over $U'_k$ then we see that these come endowed with a descent datum with respect to the \'etale covering $\{U'_k \to U'\}$ given by the third component $\gamma_{kk'}$ of the morphisms $\varphi_{kk'}$ in $\mathcal{C}$. Since $\mathcal{X}'$ is a stack this descent datum is effective and translating back we obtain a $1$-morphism $x' : U' \to \mathcal{X}'$ such that the compositions $U'_k \to U' \to \mathcal{X}'$ come equipped with isomorphisms to $x'_k$ compatible with $\gamma_{kk'}$. This means that the morphisms $\alpha_k : x \circ a_k \to x'_k \circ i_k$ glue to a morphism $\alpha : x \circ a \to x' \circ i$. Then $\xi = (U, U', a, i, x', \alpha)$ is the desired object over $U$.