What property allows ozone to absorb uv light
Shortwave UV light (100-240nm) has enough energy to knock an electron out of an $\ce{O_2}$ bond creating an excited molecule $\ce{O^{excited}_2}$ which decomposes to two free radicals of oxygen. Each of the free radicals then combines with different oxygen molecules to create a net of two ozone molecules. 

The crux here is that the electron is not "knocked out" of the oxygen molecule, the electron is just moved to a higher energy orbital of the same oxygen molecule. Since orbitals are quantized it takes a very specific energy to make the transition happen. 

Now there is a second reaction. Longwave UV light (240nm - 350nm) has enough energy to knock an electron out of an $\ce{O_3}$ bond creating an excited molecule $\ce{O^{excited}_3}$ which decomposes to a free radical of oxygen and an oxygen molecule. 

Thus oxygen protects the ground from shortwave UV, and ozone from longwave UV. Neither alone would protects us from all UV as effectively.  

Now the rest of the story is that all molecules bouncing around in the air are not in the same energy state due to atom-atom vibrations. So the UV absorption occurs as a "band" rather than as a "fine line" like in the absorption spectra of an atom in free space.