Which data sets would help to predict (exponential) market trends?

From a quick look, my guess would be that maybe they mine data from some predefined target websites, typically social media: twitter, facebook, etc. I could imagine that they capture trends in the sense of "what people are talking about" on social media platforms, but maybe they have other sources and methods.

There are Natural Language Processing methods which can be used: the basis would be some form of topic modelling, then there could be some more specific applications such as event detection. We can't know how much manual post-processing is involved though, there could be a manual filtering to remove irrelevant stuff.