Radiation in the blackbody is reflected from the walls because it's absorbed and quickly remitted by the atoms on the wall, why don't they just absorb it but reflect it totally??
That's not how the black-body system works. PLease read even the simple explanations at wikipedia or the many physics sites.
A black body (ideal) absorbs all photons, and then emits different photons in quantities according to the current temperature of the material.