Why the nach in the sentence?
The verb is "nachdenken".
That makes sense.
denken is more like to believe or to speculate and...
nachdenken is more like to contemplate or to ponder
Am I correct?
Yes, and "denken" is also simply 'to think'
Would "About what do you think?" be an acceptable answer?
It could be considered technically correct if you didn't want the preposition at the end of a sentence, but it is highly unnatural sounding.
@yoyoninja96 : I don't think that would be acceptable.
No. I don't think that that is an acceptable sentence in English.
It is an acceptable answer at Duolingo and as well as in English.
Why 'what do you think of' is not correct?
On it's own, that isn't a good English sentence. Usually you would follow 'What do you think of....' with something like 'the weather, this party, the food' - soliciting an opinion.