"Introspective" means looking inward, and there are plenty of ideas that come to mind in The Scarlet Letter by Nathaniel Hawthorne. Though I can certainly offer you some ideas to jump-start your thinking, you will have to find something that resonates (strikes a responsive chord) in your own heart or thinking.
Because this is a novel which is set in a Puritan world, the way that sin, guilt, and shame are dealt with is generally quite different than the way a modern society deals with these things. In fact, in many situations, modern society does not address these issues at all. Think about what it must have been like for Hester to have that baby in that setting, and then think of what it would be like for her today. The reaction of her town was almost certainly not what a young woman in the same circumstances today would experience. Why do you think that is? What do you think has caused that change in thinking? Which society is better and why? These are the introspective questions you might ask yourself.
Another idea to examine is the way each person chooses to deal with his sin. We see Hester boldly claiming it via the scarlet letter and Arthur punishing himself in secret for it; which is better? Is there a better way to deal with it? If so, what is it? If you have some thoughts about these things, they would be a good foundation for an introspective essay.
Compassion is in short supply in this community, as nearly everyone in town seems bent on punishment rather than forgiveness. Are compassion and forgiveness the mark of a weak society or a strong one? How do we balance sin and forgiveness, punishment and compassion? These two ideas, and how to balance them, are really important elements in any culture, and keeping them in balance is essential to a healthy citizenry. How does that work in your own life--are you a better person when someone (or God) holds you accountable for your actions but also demonstrates forgiveness and compassion? Think about how this works in your own life, and you will not have trouble writing your essay.
The Puritans did not believe that redemption was possible.
And be the stern and sad truth spoken, that the breach which guilt has once made into the human soul is never, in this mortal state, repaired. It may be watched and guarded; so that the enemy shall not force his way again into the citadel, and might even, in his subsequent assaults, select some other avenue, in preference to that where he had formerly succeeded. But there is still the ruined wall, and, near it, the stealthy tread of the foe that would win over again his unforgotten triumph.
In other words, once someone sinned he was always a sinner. (If he stole once, he was labeled as a "thief" for life; if she lied once, she was always a liar.) What is the effect, individually and collectively, of this thinking?
Finally, Puritanism (at least in this form) no longer exists. Why do you think that might be?
These are some things you might think about as they apply to your own life and way of thinking, but they are also researchable if you need some evidence or documentation. Introspection simply requires you to think about the people, events, and story in the novel through your own world view. You have to ask yourself how you feel or think about these things and why. Even the idea that revenge does more harm to the one seeking it is an idea to consider in this novel. Think, then simply turn your thinking about these things into an essay.