"How exactly should you write these types of essays?"
With Complete Sincerity!
There is nothing conceited about stating what attracts you to a certain school or job, and why you would be a great choice. When you list what you've accomplished, or your special skill set, etc. -- it's not bragging. They're just facts. The goal is to connect them to what the school or employer is looking for.
If you don't know what you want in this world, how do you go about getting it? Be yourself, be honest and confidently explain what makes you the best for the ..whatever.
It can be fun once you've practiced a bit