The tailpipe and silencer assembly is what you have pictured. The tailpipe and silencer is at the end of your exhaust system. Because of its position in the system it gets hotter and colder more often, and corrodes more quickly then any other part of the exhaust. Exhausts tend to rust from the inside to the outside first. This is what has happened, and it has released the fibre-glass sound absorbtion material from inside the assembly.
You need to replaced the tailpipe and silencer, labelled resonator in your picture. Not changing the resonator will allow the fibre-glass to shift and block the exhaust. A blocked exhaust will prevent the engine from starting if the blockage becomes great enough.