typical contexts
If we think of an (∞,1)-topos as a generalized topological space, then it being ∞-connected is the analogue of a topological space being (weakly) contractible, i.e. weak-homotopy equivalent to a point.
It is an (∞,1)-categorification of the notion of a topos being connected.
Let $\mathbf{H}$ be a ((∞,1)-sheaf-)$(\infty,1)$-topos. It therefore admits a unique geometric morphism $(L\Const\dashv\Gamma)\colon \mathbf{H}\xrightarrow{\Gamma}$ ∞Grpd given by global sections. We say that $\mathbf{H}$ is $\infty$-connected if $LConst$ is fully faithful.
More generally, we call a geometric morphism between $(\infty,1)$-toposes connected if its inverse image functor is fully faithful.
An $\infty$-connected $(\infty,1)$-topos has the shape of the point, in the sense of shape of an (∞,1)-topos.
By a basic property of adjoint (∞,1)-functors, $LConst$ being a full and faithful (∞,1)-functor is equivalent to the unit of $(LConst \dashv \Gamma)$ being an equivalence
By definition of shape of an (∞,1)-topos this means that $\mathbf{H}$ has the same shape as ∞Grpd, which is to say that it shape is represented, as a functor $\infty Grpd \to \infty Grpd$, by the terminal object $*$. Hence it has the “shape of the point”.
As in the case of connected 1-topoi, we have the following.
If an $(\infty,1)$-topos $\mathbf{H}$ is locally ∞-connected (i.e. $LConst$ has a left adjoint $\Pi$), then $\mathbf{H}$ is connected if and only if $\Pi$ preserves the terminal object.
This is just like the 1-categorical proof. On the one hand, if $\mathbf{H}$ is ∞-connected, so that $LConst$ is fully faithful, then by properties of adjoint (∞,1)-functors this implies that the counit $\Pi \circ LConst \to \Id$ is an equivalence. But $LConst$ preserves the terminal object, since it is left exact, so $\Pi(*) \simeq \Pi(LConst(*)) \simeq *$.
Conversely, suppose $\Pi(*)\simeq *$. Then any $\infty$-groupoid $A$ can be written as $A = \colim^A *$, the (∞,1)-colimit over $A$ itself of the constant diagram at the terminal object (see the details here). Since $LConst$ and $\Pi$ are both left adjoints, both preserve colimits, so we have
Therefore, the counit $\Pi \circ LConst \to \Id$ is an equivalence, so $LConst$ is fully faithful, and $\mathbf{H}$ is ∞-connected.
connected topos / ∞-connected (∞,1)-topos