I've been curious about what really happens when UV-C rays get
screened out by the ozone layer and how the wavelength is shortened
because of the ozone. Think you could help me out?

The wavelenth does not get shortened.
Ozone layer is conductive, and conductive materials absorb radiation:

IT IS IMPORTANT TO HAVE O3 IN THE STRATOSPHERE --

It is earth's primary shield against damaging uv-b radiation. This
radiation is also short wave (280 - 320 nm) and high intensity. Ozone
absorbs radiation in the 220 - 320 nm band and so keeps most of it
from penetrating to earth. Why do we want to be shielded from uv-b
radiation?
http://oregonstate.edu/~muirp/stratozo.htm

http://www.fmi.fi/research_atmosphere/atmosphere_2.html
http://www.science.sjsu.edu/scied/255/dcurley/uv_light.htm


   Energy of absorbed radiation is mostly dissipated (converted to heat)
but some is re-radiated as lower frequency (longer wavelength) radiation
by  scattering processes.