Mathematically, surface area is a squared dimension while volume is cubed. 3 squared is 9, but cubed it's 27. 4 squared is 16, but cubed is 64. Volume gets much bigger much faster because it's working in three dimensions.
The whole is thing is a catch 22. The cell wants lot of surface area in order to move stuff in and out quickly, but wants a small volume to move stuff across itself really quickly. You can't have both.
But a smaller cell has a better surface area to volume ratio, which means the difference between the total surface area and the total volume isn't that large. Good ratio = efficient cell.
Surface area is a tremendously important concept in science (particularly biology and chemistry). Larger surface area is beneficial in cells because in allows faster diffusion. Think of a cube of sugar and grains of sugar. The grains of sugar will dissolve more quickly because more liquid is able to surround the sugar than with a cube of sugar.
The reason why cells have a increasing surface area to volume ratio as they become smaller is because those two factors are not linked proportionally. Mathematically, it has been proven that larger volumes result in smaller surface areas.
As you mentioned, as the size of a cell becomes smaller, the ratio of surface area (SA) to volume (V) increases. If you calculate the surface area and volume of different sized cubes, it will show you how the numbers come about. Here is an example from ibguides.com
Surface area: 6 sides x 12 = 6 cm2
Volume: 13 = 1 cm3
Ratio = 6:1
Surface area: 6 sides x 32 = 54 cm2
Volume: 33 = 27 cm3
Ratio = 2:1
Surface area: 6 sides x 42 = 96 cm2
Volume : 43 = 64 cm3
Ratio = 1.5:1
A cell operates the best when its surface area is significantly larger than its volume. This is because SA is responsible for the transportation of nutrients into the cell and waste out of the cell. The inside of the cell uses up the materials, and so need a larger SA to keep up with the waste excretion so that it doesn't accumulate.
In summary, the surface area correlates to the rate that substances enter and exit the cell, while the volume represents the rate that substances are produced and consumed within the cell.
Surface area is very important to cell function because everything the cell needs passes through the cell membrane. As cells decrease in size so do their vollumes however, surface area decreases at a much slower rate. As a result small cells have more than enough surface area to take in the nutrients needed to support the vollume of the cell.