In a large clump of a reactant, only those atoms and molecules on the outside can collide with another reactant. Because reactions require collisions, only those atoms and molecules that are on the surface can react, since only those particles can collide with anything else.
If that clump were cut , all of the atoms and molecules along the newly created surface are now available to collide and can therefore react. So, as the particles get smaller and smaller, a larger proportion of the atoms or molecules are on the surface and can therefore react. In the extreme case, a gas (made of separate atoms or molecules) is essentially all surface. Since every atom or molecule can collide, every one can react.
Thus, the more surface area there is, the more atoms and molecules are available to collide and the rate of the reaction will increase.