Does bleach destroy RNAse activity, and if so, how does it do it?
RNAses are enzymes, and there are various ways to inactivate them. Unfortunately, RNAses are rather stable proteins and autoclaving doesn't completely kill their activity. The common methods to inactivate are unspecific methods that will destroy any enzymes, either through covalent modification or degradation.

The most common methods for RNAse inactivation are:


Treatment with DEPC. This covalently modifies histidine and a few other amino acids, which will inactivate RNAses. 
Heating to 200-250 °C for at least an hour, this is most commonly used for metal and glass equipment
Incubation with 1M NaOH for an hour at 37 °C, which will hydrolyze proteins


Using bleach is not a method I've used, but it should work in principle as hypochlorite will react with proteins.