If N gains 3 electrons from other atoms, why is it written N^-3 (with a - 3) ?

Because number of electrons becomes greater than protons and atom form anion.

Explanation:

In order to understand this problem we must know about anion and cation.

Anion:

When an atom gain the electrons anion is formed. The negative sign shows that atom gain electron because number of electron are greater than protons or we can say that negative charge becomes greater than positive charge.

Cation:

When atom lose electron cation is formed. The atom thus have positive charge because number of positive charge i. e protons are increased are greater than negative charge or electron.

In given problem N gain three electrons which means negative charge becomes greater that's why the extra electron gained by atoms are written as - 3.