How can blackbody radiation be explained by quantization?

I don't understand why quantization makes a peak in the blackbody radiation curve (so there is no UV catastrophe ) and the concept of a relationship between that peak and quantization. When the blackbody is heated, it starts to glow. All atoms start to vibrate. Total heating energy must be divided and equally shared by all atoms of the blackbody; right? But what happens next? Do some atoms get much more energy to radiate at higher frequency and waste the energy? What happens at the atomic level, so that quantization can explain it? What does classical physics claim, so that it cannot explain the radiation curve?