[morzh]

I think this cloth keeps dust and particles from getting into the space between the voice coil and the magnet pole.

In today's speakers the rippled spider protects that space but in the older speakers where spider did not go over that space it would easily get clogged.
When you remove the cloth, it is fine if you clean the speaker and keep it in the clean environment, but then there are still particles and dust, and perhaps pet hair etc etc, so I would re-glue the cloth back after cleaning.