Thanks for your comment Stelios, and my apologies for the long delay in responding!
I don't think that it would be better to do clustering on the 3-d post-t-SNE representation, as by that point we have thrown away a lot of information, and also because t-SNE is such a non-linear transformation, you would get some strange results.
As to the point of t-SNE here, the idea was to get a visual picture of what was happening within the data. The reason you see these uniformly-coloured starbursts is because that's how t-SNE represents large numbers of very similar points. In a more linear dimensionality-reduction, you would see these points all one on top of another, which would hide how many there are. When there is a greater variety of points, you get more interesting, "galactic" looking visualizations.