The Purpose Of Speaker Protective Grille

The primary function of having a grille and/or mesh in front of a speaker is for protection.
This is why you'll nearly always see these perforated shields in public address speakers, instrument amplifier cabinets, and other speakers that are regularly moved around and have a higher risk of being damaged.
For the sake of speaker longevity, we must keep the diaphragm, voice coil and the rest of the driver protected. This can be done by keeping the speaker out of harm's way or shielding it with a grille.

A speaker's acoustically transparent protective layer will typically be soft or hard. Let's discuss the Soft Mesh Grilles.

Soft speaker grilles are made from various fabrics (weaved or stitched), foam and other soft materials. We see soft speaker meshes on some guitar amps, home theatre speakers, computer speakers, and other speaker types.

Soft speaker mesh is relatively absorptive and produces fewer reflections, phase issues and resonances than its hard counterpart.
It is also freer to move along with the sound waves, thereby reducing its impedance to the sound produced by the speaker. This quality also makes soft mesh grilles less prone to rattling when the speaker produces high sound pressure levels.
The soft mesh grille may offer more or less water resistance to the overall speaker design depending on the material used. As for protection from physical trauma, the soft speaker grille is susceptible to being torn and/or stretched. Once damaged, it may not fully protect the speaker from being torn and/or stretched as well.

Do Grilles Affect The Sound Of The Speaker?
Any impedance to sound waves will affect their propagation, even if grilles are largely designed not to affect the sound of their speakers.
The perforated protective shields known as grilles and meshes do, in fact, impact the sound of their speakers. Generally speaking, the sound quality will be subjectively better when the grille is removed.