Writing for yourself is a stage in the writing process. Teachers assign journals where students can respond spontaneously to readings or films or class discussions, because journals give you space to think, ask questions, explore and discover, without worrying about a reader. We all need to write informally to develop as writers or thinkers. But formal writing is different. It may begin with a process of personal exploration and discovery, but in the end it needs to communicate to a reader.
Readers have needs that are different from the writer's.
The process you went through to discover your idea is not the same as the process the reader needs to go through to understand your idea. It takes time and experience to learn how to write for a reader. It is very hard to get distance from our own words and ideas. That is why it is so helpful to have someone else read what you've written. The feedback they provide helps you imagine your reader as you revise your essay.