1. Why do we need to be able to multiply complex numbers in polar form?

One use of the multiplication of complex numbers will be for the development of De Moivre's Theorem in the next topic. De Moivre's Theorem is then used to solve a whole range of equations.

2. Is there a quick way to remember the formulae for the multiplication and division of complex numbers in polar form?

Just remember that the modulii are multiplied or divided, and that the arguments are added or subtracted respectively.