You've written a solid essay here, and I wish you luck applying to your writing camp! You've answered the prompt, chosen a solid memory to talk about, and then expanded it by reflecting on the experience. I'll start from the beginning and comment as I read.
For starters, I think you have a good idea behind your introduction, but for an essay that you only have 300 words for, it doesn't feel necessary, especially since your audience is likely going to agree with your theme that words are in fact not powerless. In my opinion, your second and third sentence could be condensed into one, leaving you with more room to expand on your moment.
Also, in these types of essays, it can sometimes be more effective (but just as elegant) to condense some of your phrases. This, again, frees up more words for you. Here's an example:
At first glance, words can appear to be powerless.
In this sentence, "can" puts a modifier on on your statement and "to be" is just extra fluff, leaving it weaker than if you just said "At first glance, words appear powerless." And another example:
Why, then, does humanity have such a continued obsession with the power of language?
You can easily replaced the underlined portion with "continue to obsess over" and not lose the meaning and impact of the sentence, while also making your sentence flow easier and more to the point. I bring this up in such detail in the first few lines because I feel like you can easily slash a couple of words from each sentence, ultimately giving yourself more room to breathe and expand upon your ultimate point. I won't be so nitpicky moving forward, as this is your writing after all, but I feel it's a good tool to have.
I was immediately alarmed, but when I asked what was so upsetting, she said it was my poetry.
This sentence stuck out to me, because the wording feels slightly stiff. You've done a good job of maintaining a consistent tone throughout, but the phrase "what was so upsetting" doesn't sound like a question you'd ask a friend, but instead a stranger. It might be stronger to go with something more conversational, to emphasize the friendliness and the connection that was building between you two because of your poem. As well, you could easily turn the phrase "I was immediately alarmed" into a clause before the second part of the sentence, like "Alarmed, I asked what had happened." or something like that.
The conclusion of your essay felt like the strongest part to me, because you phrase the way that you feel about writing in such an eloquent but succinct way. The only thing is I think you may want to reserve the use of the word "powerful" for in the context of the power of words, as you described in the beginning, that way your message doesn't accidentally get muddled. I'm talking about in this line:
They help de-localize our pain, spread it out farther, so that it no longer becomes so pressing and powerful.
Perhaps instead of powerful you could use overbearing? Or suffocating? Then you could easily reuse the word powerful to connect your introduction to your conclusion, like whatcha suggests.
Hopefully this was helpful! I think you develop your ideas very well throughout this, and it feels like a touching piece to read. The main thing that you could work on is simply editing down some of the unnecessary filler words, and then rounding out your conclusion once more so it feels all wrapped up nicely. Great job! Let me know if you have any questions~