Writing a five-paragraph essay is not too difficult once you establish your three body points; in this case, the body points would be three reasons "why we write."
The introductory paragraph must let the readers know generally what they will be reading and set the tone of the piece. It must also have a specific topic sentence which is a preview of what will follow in the essay. The topic sentence can also be called a purpose sentence or a thesis sentence; it is one sentence which clearly states the three body points which follow it. It should capture the reader's interest, as well.
The three body paragraphs should each discuss one reason "why we write." They must be connected with clear transitions and give significant, interesting, and pertinent details about each point. The body paragraphs develop and expand each reason you listed in your purpose statement.
Finally, the conclusion should review (restate but not in the same words) the purpose statement and then leave the reader feeling satisfied. Perhaps you can start a short story in the intro and end it here, ensuring a cohesiveness throughout the essay. Other strategies include finishing with an apt quote or personal anecdote (which are also possibilities for the intro). The how-to eNotes site linked below is an excellent resource for some more specific strategies to write a persuasive essay.
Now for some ideas about "why we write." I am not certain to whom this "we" is directed or what, exactly, is meant by writing (literary writing, personal writing, or just plain ol' writing), so I will make my best guess. You can adapt these ideas to whatever the prompt means to you.
Some reasons to write include: expressing oneself, recording information, creating something, sharing experiences, and more. If the prompt means why do writers write, the writepractice link (below) gives four good reasons why writers keep writing.
1. We write to be fully alive.
We write to be fully alive. Writing draws us into the moment. We see the blades of grass, hear the miniscule [sic] chirp of the morning cricket, watch the shade travel from one edge of the yard to the other, seemingly for the first time.
Writing helps us make art out of everyday, ordinary moments.
2. We write to make a name for ourselves.
3. We write to change the world.
We write not just to change the world, but to create a new world.
4. We write to discover meaning.
Whether you use these kinds of ideas or have others of your own, they can be listed in your purpose statement and then supported in your body paragraphs by the use of details, quotes, examples, illustrations, explanations, statistics, or whatever else makes sense to you. Finish with a review of your points in the conclusion, and you will have written an organized five-paragraph essay. Happy writing!