Why is black-body radiation curve smooth without a sharp cutoff?
The key to understanding the curve is thinking about how energy is distributed among components of the system
The mathematics that produce the curve involves some not-very-simple statistical mechanics some of which Planck didn't understand when he first developed his theory.
But it isn't that hard to get the intuitive idea. Consider the case of a gas where molecules have some kinetic energy. The individual molecules don't all have the same energy: some are moving faster than others. The temperature of the gas is function of the average kinetic energy of the molecules. But individual molecules are exchanging energy very frequently as they bang into each other. But those random collisions exchange random amounts of the kinetic energy between the molecules. Sometimes one molecule will gain a great deal of energy so it ends up with far more than the average amount. But the probability of a series of collisions giving a single molecule a very, very large kinetic energy is low and the higher that energy, the lower the probability. There is no sharp cut-off just an exponentially decreasing probability of getting a higher and higher energy.
Take that picture and apply the appropriate statistical probability theory and you get the overall Planck distribution: a curve where the average kinetic energy is given by the temperature but where individual molecules have some probability of having much lower energy (with slowly decreasing probability) and others have some probability of having much higher energy (but with a sharply decreasing probability). There is no sharp cutoff at high energies just sharply decreasing probability of reaching those levels.
In reality the picture is far more complex as molecules have vibrational and electronic energy as well as kinetic energy, but these details don't matter much for the intuitive picture.