Given a space $X$, a sheaf of sets on $X$ is a functor $O(X)^{op}\to Set$ satisfying certain conditions, where $O(X)$ is the poset of open subsets of $X$ (more generally, you could replace $O(X)$ by a site). So a "sheaf of spaces (on $X$)" would just be the analogous thing where you replace $Set$ by $Top$, i.e. a functor $O(X)^{op}\to Top$ satisfying certain "sheaf" conditions. However, this is not the way Lurie uses the term, because he is working in an $(\infty,1)$-categorical setting and "space" really means "homotopy type", i.e. an object in the usual $(\infty,1)$-category of spaces. So a sheaf of spaces on $X$ is a functor of $(\infty,1)$-categories from $O(X)^{op}\to Spaces$ satisfying a higher-categorical version of the sheaf condition.