the frequency of collisions between reactant particles increases
therefore the rate of reaction increases
Note that the mean energy of the particles does not change. However, since the frequency of collisions increases (they happen more often), the frequency of successful collisions also increases.
The rates of two or more reactions can be compared using a graph of mass or volume of product formed against time. The graph shows this for two reactions.
The gradient of the line is equal to the rate of reaction. The faster reaction with the powder:
gives a steeper line
Note that the amount of product does not change when the reaction rate changes. The same amount of product is formed, but in a shorter time. Therefore, both lines on the graph finish in the same value of mass or volume, but the reaction with the higher surface area to volume ratio reaches this mass or volume sooner.
Make sure you answer questions in terms of surface area to volume ratio, rather than just surface area. This is because the surface area also depends on the mass of solid reactant used.