Depends who you ask but generally mannifold is just some structure in a (high) dimensional space having finite dimension : a line, a curve, a plane, a rock, a sphere, a ball, a cylinder, a torus, a "blob"... something like this :
If you ask a mathematician
they would say that it is a general term describing "a curve" (dimension 1) or "surface" (dimension 2), or a 3D object (dimension 3)... for any possible finite dimension $n$. A one dimensional manifold is simply a curve (line, circle...). A two dimensional manifold is simply a surface (plane, sphere, torus, cylinder...). A three dimensional manifold is a "full object" (ball, full cube, the 3D space around us...).
TO ANSWER YOUR QUESTION what he is referring is that information in this high-dimensional space can be compressed and perserved in some smaller space. (remember images when quantified live in some space that contains information about them)