What is the advantage of increasing an aircraft's horizontal tail surface area?
Increasing horizontal stabilizer area increases pitch stability and the permissible range for centre of gravity at the cost of some aerodynamic efficiency.

Stability in pitch means that the aircraft tends to maintain constant angle of attack (which roughly corresponds to maintaining constant speed with pitch). An aircraft is stable in pitch if the aft surface flies at lower angle of attack than the front surface (this applies to both conventional tails and canards).

Making the horizontal stabilizer larger does the following:


The absolute change in lift force on the horizontal stabilizer when angle of attack changes is larger, so the aircraft returns to the trimmed angle of attack faster (increased stability).
As centre of gravity moves aft, the difference in angle of attack between the surfaces decreases to keep the plane balanced, reducing pitch stability. If the horizontal stabilizer is larger, the angle of attack changes more slowly for the same change in lift, so the centre of gravity may be permitted further aft before the aircraft becomes not stable enough.
As centre of gravity moves forward, the difference of angle of attack between the surfaces increases until angle of attack of the horizontal stabilizer exceeds its stall angle of attack and control is lost. If the horizontal stabilizer is larger, it can provide more (downward!) lift before it stalls, so the centre of gravity can be permitted further forward before loosing control authority.