First, have a clear idea of what you want to write about and sum up this topic in a single statement (your thesis). You can have this idea from the very start of the essay writing process, or you can start your research into a topic or brainstorm to help you to get an idea for your essay. The best way to practice writing your thesis clearly is to write and rewrite your thesis statement and arguments until they can be clearly understood by somebody else (a friend, a colleague, etc.) who does not know much about your topic.
After you have created a clear and easily understandable thesis and arguments, you also need to be able to keep your essay writing preparation focused on that thesis and those arguments. If you are writing a research essay, concentrate your research on your thesis and arguments because otherwise you run the risk of being overwhelmed or distracted by all of the information that may be available to you. If you are writing an essay that is supposed to be more personal, then make sure that your outline is written so that you concentrate it around your thesis and arguments. The best way to practice this skill is to review your notes or outline after having taken a break from writing them. Then highlight the points that are the most relevant to your topic.
Think laterally. This kind of thinking means that instead of approaching your topic directly you approach it from a related topic or point of view. For example, if you are writing about Shakespeare's writing, then you could start your essay with a section on Chaucer's influence on Shakespeare's writing. Thinking laterally will help you to write things that your reader will not expect, and this will help to keep your reader interested in what you have to say.
Writing concisely requires you to have a careful plan in place before you start writing your essay. Having a strong, clear idea of what you want to write from point to point will help you to avoid writing sentences and sections that are not relevant to your topic. A good way to practice this skill is to plan and to subplan (that is, plan your sections, and then plan out how you will write the points that you have made for each section).
To practice closely editing your own writing, take your time while editing your papers to build a working understanding of grammar and style and read the work of others, whether this means reading published collections of essays or helping your friends, relatives or others edit their essays.