
“On” is the concept of having received a great favour, feeling grateful, and knowing that you owe somebody a lot.
In Japanese it seems like less of a subjective feeling, and more of a tangible object – a thing that you now possess because somebody helped you greatly. In Japanese, “I am very indebted” is literally: “I have a lot of “on””.
Now that you possess this “on”, you can’t ignore it. The relationship has changed because of it. You have gained some kind of duty, or obligation.
You may not have a chance to return the favour, or “pay back” the obligation immediately, but that does not matter. The point is that the relationship has changed, most likely deepened, and that is not a bad thing. By being indebted to each other we can strengthen personal connections.