Imagine that you started with a flat grid of steel mesh and each vertex in the grid is welded nice and rigid at 90 degree angles, but each rod in the grid was a kind of piston that you could lengthen or shorten. That would be a physical analog of a conformal transformation, it preserves angles but not lengths.
I would add that you should imagine being able to bend the sides as you lengthen and stretch them, as long as you keep the vertex angles frozen. Varying curvature can be introduced at each point, but the conformal requirement of conformal flatness implies that the curvature can be characterized by a single function on the manifold.
Since, in 2-D, all curvature can be described by a single function, the result is that all 2-manifolds are conformally flat.
In 4-d spacetime, conformal flatness is equivalent to vanishing Weyl tensor. In GR this means that if a vacuum region is conformally flat, it is simply flat.