Edit: I forgot to say anything about how to visualize this space. People believe it is a ball, but have not so far proven this (to my knowledge). It is the image of a map from a polytope Postnikov introduced in the paper you are reading. The restriction of this map to the interior of the polytope is a homeomorphism, but many points on the boundary of the polytope get identified with each other. My impression from talking with people who've worked with this space is that they can visualize it to varying degrees, but mainly I think try to understand it by trying to understand this map really well. Another paper you might find helpful is:
MR2525057 Reviewed Postnikov, Alexander; Speyer, David; Williams, Lauren Matching polytopes, toric geometry, and the totally non-negative Grassmannian. J. Algebraic Combin. 30 (2009), no. 2, 173–191. (Reviewer: T. Oda) 20G20 (05B35 13F60 14M25 52B70)
This proves it has a CW decomposition and relates this space you are considering to the totally nonnegative part of a toric variety. Perhaps others here will have more to say about how to visualize this space.