Why shouldn't you use first and second person pronouns in writing?
If you're writing using an academic voice (for example, an article or an essay) it's best to keep your personal opinions out of it, but there are a few exceptions.
When using academic voice, it's important not to say things like "I believe", and "I think", and "In my opinion" because the essay is not about opinions, it's about fact and analysis Of course, there are a few exceptions (like a "This I Believe" essay, or a simple reflection paper). Same with second-person. You usually don't want to address the audience directly with second person, but there is an exception when you are asking a rhetorical question (ex. "If you were in his shoes, would you do such a thing?) to illicit a specific or obvious response.
But if you are writing a fictional story, using an anecdote in your essay to emphasize your point, or things like that, it's okay to use first or second person.