All electrical machines have some amount of dis-symmetry to them, just because they are constructed of individual parts. Sometimes this is in the steel portions (segmental laminations, or unequal airgaps, etc) and sometimes in the winding (unequal resistance paths, different connections and/or lead lengths, lead routing, etc).
As soon as there is a difference in the magnetic path (or the magnetic strength), some amount of voltage will be induced on the motor shaft. The purpose of a ground brush (or wire, or strap) that makes contact with the shaft is to drain this potential away safely so personnel do not get an unexpected electrical shock.
Operating on a variable frequency drive (VFD) will increase the likelihood of having some amount of induced voltage because now the WAVEFORM itself is not purely sinusoidal at a single frequency. All those harmonics created by the switching action of the VFD result in small differences in the current wave shape - which in turn means distortion of the magnetic field, leading to shaft voltage.
The most obvious result of this is the damage that occurs to a bearing (typically either the raceway or the rolling element, for anti-friction bearings - or to the Babbitt surface for a sleeve type bearing). The damage is caused by the arcing from the shaft (which is at some voltage due to dissymmetry) and the "grounded" bearing support. The arc passes through the lubricating film, and vaporizes small bits of the contact surface. It will also carbonize most lubricating materials to some extent, leading to increased friction.
Grounding the shaft at one end is cheap insurance to protect both bearing and personnel over the long term.