Not being specific about the who, what, when, where or why is an easy way to increase the chance that your letter will just be dismissed.
Look up the exact names of the companies, entities, government agencies and people you are discussing. Don't guess or wing it. These things matter. But more importantly, you will make yourself seem more credible when you are describing people, places and things correctly!
This doesn't mean that you must write out the full name every single time, just that you must create a "defined term." For example, the first time you write about Big Bad Insurance Company, Inc. ("Big Bad"), define it by putting a shortened version inside of parenthesis, quotes and then underline.
Remember to capitalize each word (this is what alerts the reader to the fact that "Big Bad" has a defined meaning). Now every time you want talk about the Big Bad Insurance Company, Inc., you just need to write: Big Bad.
Another common mistake is not explaining the situation and/or request in a systematic manner.
At first blush, an appeals letter is meant to persuade the recipient to give you what you want (i.e., the insurance company should reverse its denial of your claim for coverage). But, sometimes more importantly, it is a PROTECTION MECHANISM intended to keep an accurate record of the interactions and history between you and the insurance company.
One day, this letter might be the only hard proof you have that you requested an appeal or spoke to a certain insurance representative or whatever your particular case may be.
This is why you should write EVERY SINGLE LETTER as if the person who is reading it has ZERO KNOWLEDGE about what is going on. Imagine that you are in court and only have this one letter to explain what happened - would a judge or jury understand what the situation is just from your letter? If not, you need to delete and try again.
If you are struggling to do this, just start from the beginning and use dates a sign posts!