It is a letter that gives the authority and the power to the holder of the letter.
For instance, If you are writing the letter to a travel company, to allow your minor child to travel to another place to see a relative, you can address it with, 'To full name '. Thus, get them done in advance in so that they are validated.
The body of the letter, will mention the reason for writing the letter. Your answer for why, will be remain the crux of the letter. Letters of consent are needed when your child has to be given a permission to travel oversees, to see a divorced parent, to allow your colleague to do something in your absence, to take of some bank deals and basically to permit someone you trust to act on your behalf.
The language of a consenting letter has to be confident, as it must show trust in the other person. As this is a formal letter, explain your reasons in the simplest language possible.