Expressing motivation in the form of a letter is not a straightforward task. Wouldn’t it be much easier if you could just call up whatever gatekeeper you need to persuade and let them know you are indeed motivated, with all the emphasis and emotion that verbal communication permits? Maybe, but personally I advise you to write down your points first anyway.

“How can I tell what I think till I see what I say?”
—E. M. Forster, Chapter V, The Plot

He had a point; writing allows us to do many things that spontaneous speech (or even planned speech) does not. It allows us to consider the validity of our premises and the persuasiveness of our evidence; it enables us to evaluate the quality of the claims we infer from them, as well as carefully appraise their relevance to our intended audience. Not only is there much more control in a motivation letter as opposed to a motivation monologue, there’s opportunities to learn new things about what you think, what you have to say, what you’ve achieved, while honing the most effective, impactful means of conveying your message to the world.