How does a high intelligence person think?

Thinking is separated in two types: the fast (intuitive) thinking and the slow (logic) thinking.

Fast thinking feels instant and works in parallel, by association.

Slow thinking feels like hard work and works serially, step by step.

This is the same for all humans, regardless of their IQ.

Higher IQ people are biologically better at making associations. Memory works by associations. So, high IQs remember more things and therefore they can associate more things.

The more things you can associate, the more ideas you come up with fast (intuitive) thinking. Hence, the higher the probability to come up with something original.

On the other hand, the more you can associate, the more ideas you can remember simultaneously. Hence, the more far you can go with logic (slow) thinking before you forget where you came from.

This is key: A lot of times you will see people unable to follow up, although the explanation offered is presented in simple steps. The problem is, there is one step too many, so people cannot hold all these steps simultaneously, they forget things and get lost. 

This is also why higher IQs learn faster: they can store more, more easily, and make faster the necessary associations between the new ideas. A normal IQ will struggle when presented with a new science, because he needs time to absorb all the new language and the associations. A higher IQ will struggle less.

That's all there is to it, I believe.