Why does the comb attract the pieces of papers if they're neutral?

This is because the neutrality of polarity can be changed by electric field in this case. When you create - charge in the comb and you expose the pieces of paper to the electric field created by the charge, you will polarise them so that the part closer to the comb will be + and the other will be -.

Here, see the electric field. The same polarities do not like each other: