How do I write a letter to my clients telling them I'm raising my hair salon rates?

In as few words as possible. Tell what is causing the increase such as cost of supplies, rent and utilities and maybe salaries to keep talented hairdressers (whichever apply).
Can you give a discount coupon just for existing customers for a few weeks or a month to show that you value your customers? Or phase in the increase maybe.
Any steps you can take to soften the blow will be valuable.
But a long drawn out letter trying to justify will just make people think of why they can't afford the increase for the same reasons you are raising rates. I am not an expert in this area.

As smear said keep it simple. Everyone is used to prices going up.

A simple:

Thank you for your patronage in the past. Unfortunately rising costs force me to raise my rates. Following is my new price schedule. I hope you will continue to patronize us.