I think I get it after watching a few more lectures. If you're moving on the surface and can't look up you have no way of knowing if the "tubes" are entangled and no way of knowing if the "dimension" you're traveling on changes even if you're going straight. A given axis is not intrinsic to the surface but made up by an omniscient observer outside of the manifold who wants to track your movement. I came up with an analogy based on how manifolds are constructed from squares in the lecture. I have an old street map, really an atlas, of all of eastern Massachusetts. They basically cut a giant map into rectangular pages. Along the edge of each page is a note that says which page to turn to if you fall off that edge. Each page is locally euclidian. For a normal atlas the entire space is euclidean because of the assumption that if you fall of the bottom edge of page 23 you will come in on the top edge of page 39. If the publisher is allowed to change how the edges are connected, shuffle all of these edge notes and out them back in different places, then he could connect the bottom edge of page 23 to the left edge of page 6 -- in either orientation.