An essay moves readers by persuading them to feel a certain way (angry, satisfied, afraid, etc.) or by activating them to do something. It is much easier to persuade someone if you genuinely feel the same feeling, and it is much easier to activate someone if you are motivated to do it too.
If something is significant enough to make you angry or afraid, maybe you should write about it. But unless you are supposed to be writing about yourself, your readers normally don't want to know why you are moved. They want to know why they should be moved. If there's something out in the world to be afraid of, your readers will appreciate knowing about it. If someone has written something that they should be angry about, you might be doing them a favor by pointing it out.
The three steps in moving the reader to feel something are (1) if you are feeling an emotion, figure out why; (2) show your readers the situation or give them the experience that has caused the emotion, emphasizing the key details; and (3) if it's not already clear, explain why that emotion is justified.
For example, if the last scene of a play felt totally unsatisfying to you, and you feel unfulfilled, (1) why? Is it because there are so many loose ends? Because the villain doesn't get the proper punishment and the hero just lets him go? Or is it because the play was meant to be performed live, but all the energy was gone when you read it by yourself? (2) Suppose it was because of the loose ends. The thesis of your essay could be, for example, that the playwright created such a rambling plot that readers should find other ways to enjoy the play. (3) You can present plot lines from Act I and Act II that have no resolution in Act V. You might point out that if the play were a comedy, it would not matter so much, because the point is to laugh as one goes along--but this is supposed to be a serious play, and yet the author has let us down. At least, perhaps, we can appreciate the dialogue.
You also can tell that an essay is effective if it activates your readers to get up and do something about what you wrote. Often, if you move the reader's emotions, you can point the reader toward an action that can express, extinguish, or deepen the emotion.
For example, if you persuade a reader to appreciate the character development in a particular book, you can point the reader to other books with a similar pattern. Or if you persuade the reader to be angry at the implications of something an author wrote, you can point out that someone else has been making those implications real--maybe the reader should go out and stop it! Maybe something that an economist wrote 50 years ago has an important bearing on how your readers should vote on a referendum in the next election. Or perhaps a moral philosopher is right that a popular practice is actually harmful, and you can persuade your readers through reason and guilt to stop doing the harmful thing.
The point of a great essay intended to move the reader is that the reader actually moves in the right direction. Maybe the reader will stop reading halfway through the essay, get up, put on a coat, and do whatever it is you recommended. If you're right and it works out, you have an A, or a revolution, or you saved civilization. If your advice is bad and the reader figures it out, beware.