This means: if for every morphism $f : X \to Y$ which is $\infty$-connected as an object of the over category$\mathbf{H}_{/Y}$ (roughly: all its homotopy fibers have vanishing homotopy groups), then the induced morphism

The $(\infty,1)$-topos $\mathbf{H}$ itself is a hypercomplete (∞,1)-topos if all its objects are hyercomplete. See there for more details.

Remarks

Hypercompleteness is a notion that appears only due to the possible unboundedness of the degree of homotopy groups in an (∞,1)-topos. The notion is empty in an (n,1)-topos for finite $n$.

An object being hypercomplete in $\mathbf{H}$ means that it regards the Whitehead theorem to be true in $\mathbf{H}$. If $\mathbf{H}$ itself is hypercomplete, then the Whitehead theorem is true in $\mathbf{H}$.