It's often seen as too casual
Consider the following phrase:

As I stated earlier, Romeo & Juliet is a tragedy.

The use of "I" in this statement implies that the author has a connection to the reader and that it's fairly casual. That's not always the case. If you were writing that on your AP Language test, the test taker isn't anybody who has met you.
Removing it takes away that implied personal connection.

As stated earlier, Romeo & Juliet is a tragedy.

It implies credibility that may not exist
The following statements have a different "feel" to them.

My name is William Shakespeare, author of the play. As I stated earlier, Romeo & Juliet is a tragedy.

My name is Michael Sexton, Director of the Public Shakespeare Initiative. As I stated earlier, Romeo & Juliet is a tragedy.

My name is Jay, an English student. As I stated earlier, Romeo & Juliet is a tragedy.

If the first two people wrote your essay, they could probably get away with using "I" because they have the credibility to back it up. But you as an English student don't (yet). Removing "I" statements means the focus is no longer on yourself and your thoughts, but rather on the arguments to back it up.
Use of "I" often a symptom of underlying grammatical or presentational problems
Using the word "I" in and of itself can be very effective, but at the high school level, it's often indicative of other problems.
Using a weak statement instead of an authoritative one
Consider the following phrase:

I conclude that Romeo and Juliet is a tragic story.

This is a weak statement. You might imagine it being followed with "but I could be wrong". Removing that phrase is more authoritative.

Romeo and Juliet is a tragic story.

Stating your opinion, rather than relying on sources
If you find yourself using the word "I", it might mean that you are relying on your own research instead of that of others.

I conclude Romeo and Juliet is a tragic story.

If you were William Shakespeare, that would be enough because of your credibility. But you aren't, meaning that this is just your unsourced opinion. The following phrase is better.

Dozens of literary critics have concluded that Romeo and Juliet as a tragic story. [citation]

Injecting feelings or subjectivity into intellectual, objective arguments
A lot of high schoolers use "I" in conjunction with feeling phrases.

I believe that Romeo and Juliet is a tragic story.

I think that Romeo and Juliet is a tragic story.

I would argue that Romeo and Juliet is a tragic story.

All of these are problematic because they rely on emotions and opinions, rather than facts, and they lack authority for that reason.
Turning an objective statement into a subjective one
As parodied by the webcomic XKCD, sometimes things are objectively true.

The Gateway Arch is the most recognizable arch in St. Louis.

Using an "I" statement can weaken that and make it subjective.

I believe that the Gateway Arch is the most recognizable arch in St. Louis.

If Romeo & Juliet meets the criteria of a tragedy, just say it. Chances are that "I" statements like "I conclude that..." are going to weaken that argument and make it seem more subjective.
Granted, there are times where a statement should be subjective, but they are rarer in formal essays and there are other ways to express that without using the word "I".
Finally, it doesn't match with conventions used in many types of writing
The answer may boil down to "because other people don't do it."
Do your textbooks use the first person when describing concepts? Do newspaper articles (aside from opinion pieces) do it? Part of writing is following the conventions of the type you are using, because that's what the user expects. As you may have noticed, Stack Exchange has conventions too, like not saying "Thanks! -Jay" at the end of your questions.
Part of the conventions of analytical academic writing is that the first person is rarely used. For instance, this article from the Journal of Palestine Studies examines media portrayal of Israelis and Palestinians in modern culture. Not once in the entire article does it use "I" or "we". This is typical in the vast majority of academic articles I've read.
Those who say that you should stop using "I" because your teacher says so are probably alluding to this reason. The convention your teacher has established in his or her class is that formal essays do not use the word "I". Hopefully, those conventions aren't arbitrary and are reflective of real-world usage, but regardless, that's the convention established. Going against it can be jarring to what your readers expect.

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