For instance, in a typical job application, the problem is that the company or organization needs a position filled, and more importantly they need someone who can do the job well and help the organization achieve its mission (which may be more profits, or it may be to save children in Darfur or to end global warming or to cure cancer). You arouse their interest by telling them that you’re the right person for the position and by promising to help them achieve their mission.
If your intent is to persuade the recipient in some way, whether it's to invest money, give you a reference, hire you, partner with you, or fix an issue, create a compelling case for your cause. If, for example, you want the reader to sponsor a charity event, identify any overlap with their company’s philanthropic goals. Convince the reader that helping you would be mutually beneficial, and you will increase your chances of winning their support. Keep your closing paragraph to two sentences. Simply reiterate your reason for writing and thank the reader for considering your request.