From the article: "In the case of the spinal cord injury data, Ayasdi’s TDA-driven approach mostly confirmed what researchers already knew: The drugs didn’t work."
How this "Ayasdi" company's analysis probably works (based on "Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival" and the original "Mapper" paper "Topological Methods for the Analysis of High Dimensional
Data Sets and 3D Object Recognition"): They take point cloud data and connect each point with its neighbors (the distance metric that is used is probably domain-specific) to build a proximity graph that approximates a simplicial complex. As input to their algorithm, they also have one or more scalar functions defined on the point cloud data that contain information which is relative to the problem at hand. For example, each point could be a gene, and maybe the scalar function value at that gene could be probability of association with some disease, and the distance between two genes might be the Levenshtein distance between their genetic codes.
With data in this form, they approximate the Reeb graph of one of the scalar functions, which is a sort of "data skeleton." They can do potentially interesting/useful things with it.
The approximation of the Reeb graph reveals zero-cycles (connected components of the simplicial complex) and some one-cycles (handles/tunnels in the graph, sort of like holes in a donut). This "skeleton" of the data allows them to do a variety of things, such as segment the data into components that are (approximately) topologically "simple" (they do not contain any 1-cycles), identify local maxima/minima, find saddle points where forks in the data merge together, and locate "essential saddles" which constitute the high points and low points of handles/tunnels. They can also remove "topological noise", which helps them to separate spurious topological thingies from features that might be important.
Their technique doesn't necessarily recover "true" topological information since a lot of what they do is approximate. There are actually more accurate techniques (e.g., simplicial homology, or fast Reeb graph algorithms) for getting an exact answer, albeit with potentially higher computational cost.
Topological data analysis is a big field, and this Ayasdi company appears to mainly use this one approach (but I could be wrong). I think they are trying to lay claim to the term "topological data analysis" and get people with money excited about it.