Too many applicants get tongue-tied at the first sentence of their admissions essays. Just how do you start? Worry about the beginning later. You don't have to start writing at the beginning of your essay and work your way through to the end. Begin anywhere it feels right. In fact, I tell my own students to begin with a bullet-point list of the points to be made and then construct the essay around those points.
Consider these starter questions.
Since the admissions essay is your opportunity to stand out, you might begin by talking about the one thing that makes you different from all of the other applicants. Or you might begin by talking about an experience that was important to you. How did you become interested in your discipline? How did you know that you wanted to go to graduate school? Write about some of your activities: Why did you start them and what motivates you about them? Engage in the self exploration you need to know yourself - and share that knowledge with the graduate admissions committee.
Take a process-oriented perspective.
Don't feel wedded to what you write, but consider writing as a process in which you can discover what you'd like to include in your essay. Write multiple drafts and you'll find improvement with each draft.
Get feedback on your writing from as many people as you can. Also consider getting feedback before your essay is finished so that you can correct and revise it before you've spent too much time on it. Your reader might see potential in an idea that you were about to trash -- and might help you to stay on track. When you think you're done, give yourself time away from your essay and you'll come back to it with a fresh eye, able to evaluate it more objectively.
Brag a little.
When your essay is complete, reread it to be sure that it discusses your strengths. It's sometimes hard for us to talk about ourselves as we're often taught that modesty is a virtue -- but it isn't when you're applying to graduate school. It's ok to feel like you're being boastful or even cheesy, as long as what you've written is true. Your goal in writing your personal statement should be to excite and inspire your reader -- and get a ticket to graduate school.