Need some pointers for starting an application essay
In general you should start writing "in any way that works". There is no connection between how you start writing and what is at the beginning of a completed written piece. As an example, many writers have written entire books, been dissatisfied with the beginning, and simply removed the first chapter entirely. Start with the part you feel comfortable talking about, the conclusion, whatever.

"Throat clearing" is a common term for the need to start with rather uninspiring junk. But if you try not to write it in the first place, you're likely to feel blocked and have trouble writing anything at all. Go ahead and start with trite, non-suggested phrases like "It's been my dream to..." or "I've always wanted to...". The process of writing is not just about putting words in print, but a process of developing and clarifying ideas. What you want to say, and how you say it, should change naturally through stages of revision - you don't have to figure out the best way to say something before you write anything at all! Get something out, and then review and revise as needed.

There is also a huge problem with word limits, in that the natural impulse for nearly everyone is to try to figure out how to express themselves in so few words before they've written anything at all. Some people are just naturally pithy, but for the rest of us this is just does not work. Do not write to a word limit - in fact, you should naturally expect to write 2X-10X+ as much as is the strict limit for something short like this.

Nailing the word limit is then process of revision: drop parts that don't really add to your application, remove parts that are just bad and you shouldn't mention (too much self-disclosure, complaints, etc). Then, if the word limit is really small, you begin squeezing out little bits of space: trim, remove unnecessary words, find more compact and expressive way to word things. If you still have too many words, start looking for weaker statements to yank out, and pair down to the best points you make out of everything you say - even if you end up only being able to make 2-3 good points, that's better than 10 items in unsupported list format.

I am told that the more you go through this process, the more automatic it becomes, but I am skeptical that this is true. In 10+ years of writing I still can't say things in the a succinct, concise, pithy way on the first (or second or third) try. Even in this answer I've gone back and erased more than I've written out here!

Once you get out what you want to say, even if it isn't very good and is way too long, there are lots of resources you can use to refine your work, including examples, tips and tricks, asking for people to read your work and give feedback, reading your writing out loud to someone, etc. But none of that works if you don't have raw written material to start with - so start where it's the easiest for you to start writing, and everything else will flow from there.