This is a phrase that people use at the beginning of a very formal letter or e-mail. For example, if you're writing a letter to inform an employee that you are firing them, it might start:
I am writing to inform you that we will not be renewing your contract for 2011.
In addition to "that...", you can also say "I am writing to inform you of (something)". For example, if you want to quit your graduate school program, you could write to your academic advisor:
Dear Professor. Kind,
I am writing to inform you of my decision to withdraw from the program.
As you may be able to tell, "I'm writing to inform you..." is often used for formally announcing some kind of bad news.