Tell an interesting story. Of course it should be sincere, but don't feel the need to write an autobiographical essay. Like any good piece of writing, it should achieve its aims--namely, presenting the reader with a favorable impression of the applicant, in a way that pleases and intrigues. Reading a whole slew of crappy essays that all boil down to "I want to be a vet because I've always wanted to be a vet and I love animals" is boring.
One reason they have you write this essay is to see if you can communicate effectively with other people and sell yourself when necessary, be it to a client or prospective employer--this is a necessary skill for a successful veterinarian, and, above all else, the admissions committee wants to applicants who be successful both in school and their professional careers. Keep this in mind while writing your personal statement.
When I was writing mine, I made a mental list of all my attributes that the admissions committee might find desirable, then I thought about experiences I've had that I could use to demonstrate these. And then I wrote a story about it that I thought they might like to read.
And then I rewrote it and rewrote it and rewrote it, multiple drafts, then gave it to a number of people to read and comment on, then revised some more...
REVISION IS THE MOST IMPORTANT PART OF WRITING YOUR ESSAY!!!
