What is the intuition behind using LSTM for classification tasks?
I'll go for you questions one by one:


  What is the rationale behind using it in classification tasks?


LSTM Networks for classifcation tasks are mainly due to NLP. In particular, Sentiment Analysis. What is the sentiment/polarization of a tweet? By "understanding" natural language, a model can guess whether it is positive or negative towards a given issue.
Since the sequence of words is fundamental in order to understand natural language, you need to capture the sequence of words in order to understand the text, that you then need to then estimate the sentiment.

--


  Can the classification problem be reduced to sequence prediction
  problem in some way?


I think NLP made the two tasks very related to each other, even though in my opinion they can still be analytically separated, so I wouldn't say that one can be reduced to the other.

However let's say that, once you have implemented a model that is very good at estimating the sentiment of a text, you can then make it generate positive/negative texts. In other words, they are closely related (thanks to NLP) but they are not the same.

Hope this helps, otherwise let me know.