I saw the following statement in a paper of Bhatt-Mathew:

Let $X$ be a quasicompact quasiseparated scheme. Then there is an equivalence of categories between constructible étale sheaves (of sets) on $X$ and algebraic spaces $\mathcal{Y}\to X$ over $X$ whose structure map is étale.

I'm familiar with the proof that lcc étale sheaves of sets $F$ are equivalent to finite étale schemes $Y\to X$, which works by showing that constant finite sheaves of sets on $X$ correspond to trivial finite étale surjections over $X$ then using a descent argument (namely, if $F$ is the constant étale sheaf with value $S$ on $X$, it is representable by the étale $X$-scheme $X\times S$. Then the locally constant constructible case follows by unwinding the sheaf data to glue the $U_i \times S_i$ where the $U_i$ form a cover of $X$ such that the restriction of $F$ to $U_i$ is the constant sheaf valued in the set $S_i$.

For the case of an algebraic space $\mathcal{Y}\to X$ étale over $X$, I don't understand why there is a stratification $\{X_i\}$ of $X$ such that the pullback $Y_i:=\mathcal{Y}\times_X X_i \to X_i$ to each stratum is finite étale, and conversely, given a sheaf that is constructible with respect to a particular stratification $\{X_i\}$ of $X$, I don't see why this helps us build an algebraic space from the corresponding family of finite étale schemes $Y_i \to X_i$.