In cylindrical grinding the workpiece is held between centres (or in a chuck) and rotated slowly while a fast-spinning abrasive wheel removes material. A steady rest (also called a work rest or back rest) is an auxiliary support fitted between the two centres. Its job is purely mechanical support.

Why it is needed:

• Long, slender shafts have a low bending stiffness, so they sag under their own weight and deflect away from the grinding wheel under the radial grinding force.
• This deflection produces chatter, taper, barrel-shape and out-of-roundness, ruining size and surface finish.
• The steady rest bears against the workpiece (usually with shoes or rollers) at one or more points along its length, keeping it concentric and rigid, so the wheel can grind a true cylinder.

Therefore the correct reason is to support long workpieces during rotation.

The idea that it cools the workpiece is wrong — cooling is done by cutting/grinding fluid, not by a mechanical rest. Aligning the grinding wheel to the table is a machine-setup/dressing function, unrelated to a rest. Holding abrasive material is the role of the grinding wheel or a bonded abrasive stick, not a steady rest, which never carries abrasive.