The latent representations of the data are like points on a surface. That surface is the manifold. We don't typically have the full manifold and can only sample points from it by embedding data into it.

Worth noting a different manifold "exists" after each transformation (e.g. layer). You only sample from the same manifold when you apply the same transformation(s).

Also worth noting that in reality manifolds will be "spiky" in very high dimension, so the idea of a "surface" is best understood through patterns of distance between samples in embedding space and way they collapse in low D.