Intrinsic versus extrinsic dimensionality. Every point on the left is stored as three coordinates, yet two numbers are enough to say where it sits on the surface. The surface is two-dimensional on the inside and only lives in three dimensions.
Neighborhoods, geodesic distances, and embeddings. On the Swiss roll, two turns can be close through the air and far apart along the sheet. The distance along the sheet is the geodesic distance. An embedding gives every point new 2D coordinates, and the question is which kind of closeness survives. Hover a point to ring the same point in the other view, and use the shared color to see whether neighbors stay together.
Why nonlinear structure can be lost in a simple projection. The MDS-style mapping is a fixed linear projection, so it can only show a shadow of the shape. Parts that overlap in the shadow are drawn on top of each other and their colors mix. The angle-based mappings pull the surface open instead, at the price of a seam where the angle wraps around.