"Does everything have to be explicitly stated? Is it bad to imply sometimes?"
To answer this question, let's look at how Ira Glass from This American Life divides a story into two parts: anecdote and reflection. Anecdote is the stuff that happened, and reflection is why you think that stuff matters. Whether you're describing anecdote or reflection, you want to be explicit. That is, you want to be detailed. For the anecdote, that means appealing to the five senses when describing the events. For the reflection, that means making sure you answer the "So what?" question so that the reader knows why he should care. So yes, be explicit.
The problem is that explicit is not always interesting. Think about the essays you write for English class. You start with a thesis statement, which is some argument about what something means. Then you follow up with evidence that supports your thesis. It's explicit, maybe scholarly, but not suspenseful. We all know the formula: thesis, then evidence.
In the application essay, your anecdote - the stuff that happened - is your evidence, and your reflection - why that stuff matters - is your thesis. If you write the application essay the way you write a normal English class essay, you will start with the reflection and follow with the anecdote. Thesis, then evidence.
But a reflection-anecdote structure kills all the suspense. If you open with your reflection ("Activity X taught me Ability Y"), then we all know you'll spend the rest of the essay sharing some examples that support your reflection. If your reflection is "Soccer taught me patience," then the rest of your essay will be some experiences you had playing soccer. Do I need to read the rest? I certainly don't want to.
To make the reader want to read more, you need to invert the reflection-anecdote structure. Instead of writing the reflection first, write the anecdote first. That way, you imply your main point because the reader has to wonder, at least for a few sentences, "Why does this matter?" Then you resolve the suspense by stating your reflection. For example, you might open with a scene from your soccer experiences and then, after letting the reader see what's happened, start to explain why it matters to you.
So, to answer the question, be explicit in describing both anecdote and reflection. But to maximize suspense, forget about your normal thesis-evidence structure. Forget about making everything clear in the beginning. Instead, turn the normal English essay upside-down by writing an anecdote-reflection structure. Start with the stuff that happened, and then explain what it means. That's how you write an essay that's detailed but suspenseful.
Good luck writing!
Jon Perkins holds a B.A. in English from Stanford University and a J.D. from Harvard Law School. He helps students with their college, law school, and medical school applications.