The following is from an email to one of my undergraduate students in response for feedback on her application to medical school. I am not an expert on applying to medical school. These are just some thoughts stemming from lessons I've learned in grant writing that may apply. I'm posting this online just in case it is useful to others.
Dear ____________,
I think these letters are a bit like writing a grant. The first rule of writing a grant is to understand what the mission (or goals) of the granting agency is and how your proposal fulfills their mission. Similarly here you need to understand exactly the mission of the medical school -- I assume that means training good doctors, but you can call them and ask them directly or look online. (You would want to deal with the admissions people on this since they make the decisions.)
Once you understand their mission, your VERY FIRST SENTENCE should state how your acceptance into the program will fulfill that mission. (You want to state this in a nuanced way, not as a raw "You should accept me because...") Avoid vague generalities and be careful with regard to stories about your life except when they directly communicate to the admissions committee why your admittance will fulfill their institutional mission.
Salesmanship is an art. It is not hard however. It mostly means adding very direct sentences at the beginning and the end of paragraphs. Open the paragraph with a general statement about how the following sentences relate to your ability to fulfill the mission of the organization. Write the next few sentences (eg events from your life and so on). Finish with a sentence that clearly explains why these events / experiences will allow you to fulfill the institution's mission.
These are professional business people so it does not matter whether they like you or not. They are simply looking to see if by accepting you, they will have done their job of supporting their institution's mission. Your letter should to the extent possible make it an obvious choice.
Hope that helps.
THK