When you’re indebted to someone, you owe that person something. It could be that you owe money, or it could be you’re just grateful for a nice thing the person did.
You probably notice a big chunk of the word indebted is made up of the words “in debt.” If you’re indebted to someone you are indeed in debt — for either money or gratitude. You might be indebted to your credit card company, meaning you owe them money. Then you might be indebted to your mom when she gives you a small loan to pay the bill, meaning you owe her gratitude for bailing you out.