This isn't a complete answer, but consider a series of data points describing a line. E.g., (0,0,0,0), (1,1,1,1), (2,2,2,2), .... It's possible to have lines even in any n-dimensional space, but the data is "high dimensional" -- you're using a large number of variables to describe a fairly simple process.

Contrast that example with any sufficiently large, uniform sampling of the n-dimensional unit cube. The data is inherently high dimensional, and any attempt (attempts from a certain class of allowed methods -- no need to burden ourselves with the details) to reduce from n to k<n dimensions will throw away some important information about the structure of the data.

Interestingly, it's very possible to have a low dimensional representation of a high dimensional process. One of the other comments mentioned the example of a photograph representing a 3D scene (also containing a week, implicit view into some other variables like temperature). That transformation from 3D->2D is inherently lossy, but for some kinds of problems a finite sampling of a low dimensional representation allows you to uniquely reconstruct a high dimensional data representation. I haven't read the article yet, but the other comments seem to indicate something of that flavor happening here.

Disclaimer: This is reaching the limits of my math abilities.

The differences comes in the fact that higher dimensional tuples may contain data that is independent of other fields. Say, if you have a tuple that's: (name, dob, address,) and you have a projection function that accepts such a 3-tuple and returns a 2-tuple of (name, dob,). For that function, the address dimension has no relationship at all to the other fields, meaning, that there is not unproject function that a person could create such that 3_tuple == unproject(project(3_tuple)).

With manifolds, the higher dimensions can have a relationship with lower dimensional data, and such a relationship can be encoded into a function. What the research appear to have designed is a system that, given a priori knowledge of task and enough n-tuples for learning, can produce and approximation of such an unproject function.

Thus, after learning, they have a system where, unproject(project(n_tuple)) ~= n+1_tuple. Because they were able to inform the learning system about the nature of the relationship between the two dimensions.