Essays allow you to project personal traits that have contributed to your success till date and showcase potential to succeed in your future career plans.
If you have done well in your academics so far, what personal traits allowed you to perform well consistently? If you have had a successful career so far, what has allowed you to outperform others? If you are effective in out of work activities, what drives you to balance personal and professional involvements?
Well written essays project personal attributes that allowed you to succeed in the past. They also project you as a person who understands her strengths and abilities, an essential quality found in people who are consistently successful.
Essays can be helpful only if you have past successes to talk about. If you have not had much success in academic and professional areas, essays can do little to improve your chances of admission.
Well written essays can can do the trick for those who have have mixed profiles: average academic profile with a good career graph, great academic profile with average career graph, good academic and career profile with little extracurricular involvement.
Essays can be instrumental in admissions for those belonging to the competitive applicant pools: consulting, investment banking and IT. They allow you to differentiate your application from others with similar backgrounds.
To make your essays effective you have to do the grind. Spend time introspecting. Know your real strengths and motivations. Don't confuse your strengths with skills you aspire to be good at. Know the difference between your real strengths and those that you are still trying to build. Know how these strengths developed.
Think about your achievements; identify personal traits behind the achievements. Spend time alone thinking about your real strengths. Talk to friends and colleagues. Honestly assess yourself and objectively evaluate the feedback you receive from others. All this can take time. Start now, don't leave this exercise till the very end. The effort you make now will show in your essays and in the outcome of your applications.