What does the long curve-shape t-SNE mean?
The lines therefore are the time series. t-sne is a very locally sensitive algorithm so every data point is very likely to show up nearest (in 2-D) to its nearest (N-D) neighbour. 

In MNIST there is no sequence, so it doesn't show up as a line in 2-D. Every digit is written in a set of similar ways that cluster together. In your time-series data, if each time point is most similar to its next time point, what we expect is for them to be in some sequence - because you'd want point t=n to be closest to points t=n-1 and t=n+1  (if they are in fact the most similar).