Some people are very reserved when they are in the company of others. When you tell a person to let his hair down, what you mean is that you want him to relax and start enjoying himself. You want him to speak his mind.
Example: It was only after the completion of his last exam that Chetan let his hair down.
In the old days, women had to put their hair up in a bun during the daytime. The only time when they could literally let their hair down and be themselves was when they were ready to go to bed.