Explain why the mole is used as a counting unit for mass

It isn't.  

Explanation:

It is a counting unit for "things" — atoms, molecules, ions, etc. —  and these things all have different masses.

For example, a chemist may know that one atom of Cu reacts with one atom of S to form one molecule of CuS.

Cu + S ⟶ CuS

She knows that 1 mol of Cu (6.022 × 10²³ atoms of Cu) has a mass of 63.55 g and 1 mol of S (6.022 × 10²³ atoms of S) has a mass of 32.06 g.

By measuring  out 63.55 g of Cu and 32.06 g of S, she knows that there are just enough atoms of each element to react with each other.

Thus, the mole has enabled her to count atoms.