When a charged comb is brought near a small piece of paper, it attracts the piece because of induction. Induction is the process of redistribution of the charges on one body due to the presence of a charged body. Similarly, the charges are distributed on the paper from the comb. Thus when a charged comb is brought near a piece, the charges at the pointing end of the comb attracts the opposite charge on the piece of the paper which were introduced due to induction.
Whole paper does not become charged, just the area near the charged comb contains induced electrons.

Hence When a charged comb is brought near a small piece of paper, it attracts the piece.