It’s because you have edges with more than two polygons 
Enter edit on the mesh, vertex or edge mode, and use “select non manifold operation”. (You can uncheck “boundary” in operator box because that condition doesn’t matter too much.) See everything that gets selected? All of that is bad geometry, and it’s going to bite you over and over again. (In particular, here, it makes it impossible to surface deform. But it’s bad in so, so many other ways too.)
One of the qualities that makes a mesh “manifold” is that each edge connects two and only two faces. If you look at the edges that get selected, they’re getting selected because they’re shared edges-- they actually connect four different faces. This is basically going to screw up anything you ever want to do with the mesh (including, render it with smooth normals or subdivide it.)