Show don’t tell.
You want to scream every time your English teach utters this elusive phrase, I know. But basically, just bear in mind the difference between saying, "I love math," and saying, "As I flipped open my new Calculus text, the butterflies in my stomach overwhelmed me with the desire to grab the nearest pencil and graphic calculator, to submit to exploring the concepts of limits and derivatives for the night’s remaining hours." One of those statements is vastly more convincing (albeit immensely more melodramatic in this case) than the other. Focus on specificity and vivid details – captivate your readers with tributes to all five senses rather than vague references and empty declarations.
Be conversational rather than formal.
Admissions committees are interested in your voice and how you sound, so you can drop the stuffy act that you normally put on for your English and History teachers in the essays you submit to them. Be relaxed and casual – write as you would speak. However, don’t get too comfortable. Keep it within reason and be appropriate – no cursing or vulgarities.
Have others review your writing.
Another way to get around being too close to your written work, and offer it the criticism it deserves is, to have another party that’s not so close to the material to read it and offer you feedback. Consider all feedback thoughtfully – it all comes from somewhere valid – but do bear in mind that you do not have to take every single suggestion someone put forth. The more people you show your work to, the more points of view you’ll get, and that’s great – but be careful not to overwhelm yourself with too many varying perspectives. Remember that it’s you who’s making the final call when it comes to content and structure. But certainly you’d better listen when told you’ve missed a comma or spelled "opportunity" as "opportunity" instead.