I'm not too thrilled with arbitrary rules, especially in writing. One reason, of course, is that when I make a rule, I have to enforce it. Another is that poor teachers love rules so much that students become convinced that rules are the only point to English class.
So why have a rule about minimum length of an essay? If I ask for, say, a 1000-word narrative about a turning point in your life, what do you do if you only have 500 words in your head? Can't 500 words be good? Isn't there a danger that the student will write 500 words of quality stuff surrounded by 500 words of trash?
Well, yes. And you'll notice that the really good professional writers seem to get things done very efficiently, concisely, and briefly.
In spite of all this, I've got a couple of reasons for the word count.
Developing an Idea
Five hundred words, for many people, is just about enough to say the obvious material that everyone knew anyhow. It's the sort of essay you can throw together in one draft, taking perhaps an hour. You don't learn to explore an idea deeply in one page. You don't learn to answer the question, "What else?" There's no room for examples, for discussion of the other side, for developing details. There's no room for "Show, don't just tell."
Learning Structure
A really short essay doesn't need much structure. Sure, in high school you wrote five-paragraph essays that were 500 words, but they really didn't need an introduction or a conclusion very much. You often got by with a single-sentence lead and a single-sentence wrap. If the teacher hadn't told you how many paragraphs to write, you would have done the whole thing in a single paragraph. At 1000 words, you can't make the whole thing a single block—the reader gets mired down. You must have a structure.
Learning Correctness
If you really don't know how to form sentences, you might squeak by writing a page and a half. Three pages, though, gives you enough space to demonstrate that you really don't know grammar. Or that you do.
The abuses
I always know who wants to write and who is simply laying down words in hope of getting out of this class. The tricks are so obvious: large type, enormous margins, 2½ line spacing, two inch indents for paragraph beginnings (with an extra line skipped between paragraphs), and a full four-line MLA header on each page instead of just the first one. Is that writing? Why is that good? It only profits the paper company. When I receive a paper like that, I sigh because the student assumed I was too stupid to see the tricks, then I run the whole thing through the scanner.
Last semester I received several really undersized papers. The final assignment was a five to seven page research paper. One student gave me less than three. Was his work a model of brilliant concision? Did it resemble the writing of the best newspaper feature writer? Nope. It was obviously thrown together in less than an hour. It said nothing of any value and the student learned nothing from it.
Help
So what do you do when you realize that you aren't going to fulfill the minimum length requirements? If you aren't too arrogant to accept help, it's available. Every school has some form of writing lab. The textbook is usually full of advice on developing a topic. All writing teachers keep office hours. You just have to get up the humility (or courage) to ask.