A (smooth) simplicial manifold is a simplicial object in the category Diff of (smooth) manifold.
The role of simplicial manifolds in most of the literature is best understood by thinking of as a subcategory of that of simplicial presheaves on Diff. Using the local model structure on simplicial presheaves which presentable (∞,1)-category the (∞,1)-topos of ∞-stacks, one is entitled to think of any simplicial manifold as representing a Lie ∞-groupoid.
In fact, more is true: as discussed there in detail, in the projective local model structure on simplicial presheaves on Diff there is a cofibrant resolution functor with values in simplicial manifolds. (Here we allow Diff to be closed under coproducts indexed by sets; that is, we do not require our manifolds to be connected or second-countable, although we may require their components to be second-countable). Therefore every ∞-stack on Diff may be presented by a simplicial manifold.