What happens when you square volatility? Why is this important in derivatives pricing?

When you square a volatility you get a smaller number in units of volatility squared. The square of volatility appears in the expressions d1 and d2 in the Black-Scholes pricing formulas for European puts and calls.

For example, the square of 5% is 0.25%. The square of 9% (this looks like an FX vol) is 0.81%. From something small, we get something very small.