A tab is like an additional, miniature control surface, with its own change in lift. Since it sits near the trailing edge of a larger control surface, this lift change affects the hinge moment of the larger surface.
There are several types of tabs:
The simplest is a strip of metal which can be bent on the ground so it changes the deflection angle at which the control surface has zero moment.
Next on the list is a tab with can be set by the pilot, but its angle relative to the control surface does not change with the deflection of the latter. In both cases, the effectiveness of the control surface is unchanged.
More complex tabs do change their angle with control surface deflection. Flettner or servo tabs move opposite to the deflection of the main control surface and greatly reduce the hinge moment, at the cost of a small reduction in control power (think 90% force reduction for a loss of 20% in control power). On small airplanes you might even find tabs which increase the control force by being geared in the opposite way.
Flettner tab (picture source)
But it gets even more interesting with spring tabs: Here the gearing changes with tab hinge moment, so at high dynamic pressure the gearing is higher and control forces do not grow with the square of speed any more. Of course, more force reduction means less control power, but without such a device the control forces would be too high for effective control.
Spring tab, picture from this answer.
Spring tabs need careful design: Their feedback might cause flutter at specific speeds. Also, the reduction of control power from tab deflection can be as high as two thirds of the control power of the original control surface, but they could reduce the control forces down to 4% or even 2%. They were the last resort before the introduction of hydraulic controls to make large, fast airplanes controllable at high speed. Due to their high effectiveness, they require high precision in manufacturing and careful adjustment in flight testing. The Junkers Ju-88 used a combination of tabs, overhanging balance and adjustable hinge position for very light aileron forces even at 500 km/h, but each airplane required several flight hours of testing and adjustment before delivery.
how does the use of this tab differ from making the whole elevator smaller?
A size reduction will reduce control power and forces in equal measure while a well-designed tab will cost between 10% and 70% in effectivity for a force reduction of up to 98%.