Diagnosis
Dust and skin oils are getting trapped in the fine mesh fabric. You can't use liquid cleaners or scrub it, or you'll drive grime into the speakers behind it.
How to Fix It
The best tool is a soft-bristled detailing brush, like the kind for cleaning car interiors or a clean, dry makeup brush. Gently brush the fabric in one direction to lift the dust out. Follow up with the low-power suction from a vacuum cleaner hose, holding the nozzle about an inch away from the fabric. Do NOT press the vacuum directly onto it.
If That Didn't Work
For sticky spots, slightly dampen a microfiber cloth with only water—wring it out so it's barely damp. Gently dab (don't wipe) the spot. Let it air dry completely before turning the unit back on.
Heads Up
Never use compressed air cans. The forced air can push dust and moisture past the grill and directly onto the speaker drivers, which can cause damage or mold over time.