The smaller the cell the greater its surface-to-volume ratio?
This is a basic property of geometry: for any given shape, area will be generally proportional to the square of the length, and volume will be proportional to its cube. (this isn't true for all shapes by any means; the "generally" refers to the principle that length, area and volume are 1, 2 and 3-dimensional properties respectively, not that all or even most shapes behave this way... although a lot of the shapes that happen to be common in the Universe do. See end of the answer).

For example, if a sphere has a radius of R, its surface will have an area of 4*Pi*R², and its interior will have a volume of 4/3*Pi*R³. This means its surface-to-volume ratio will be 3/R, which is inversely proportional to R, meaning when R increases it will decrease and vice-versa.

Like so:

\begin{array} {|l|l|}
\hline
R~(\sim length) &amp; R^2~(\sim surface~area) &amp; R^3~(\sim volume) &amp;  1/R~(\sim surface~area / volume) \\
\hline
1 &amp;   1 &amp;   1 &amp; 1~(= 1/1) \\
\hline
2 &amp;   4 &amp;   8 &amp; 1/2~(= 4/8)\\
\hline
3 &amp;   9 &amp;  27 &amp; 1/3~(=9/27)\\
\hline
4 &amp;  16 &amp;  64 &amp; 1/4~(=16/64)\\
\hline
5 &amp;  25 &amp; 125 &amp; 1/5~(=25/125) \\
\hline
\end{array}

Put another way, if your shape is growing in 3D then whenever you add a bit of length you'll tend to add much more area (because area is increasing along two directions of length) but you'll be adding even more volume (because volume is increasing along three directions of length), and more volume than area means the area-to-volume ratio becomes smaller.

It helps that cells are convex and can often be approximated as spheres, so the rule absolutely applies to them. It's different with squiggly shapes, or cases where you aren't growing along every axis (for example this principle is true if you increase a cylinder's radius and length, but not if you just increase its length; at that point length, surface area and volume are all increasing along a single dimension and thus all at the same rate; check the equations to see this). You often do find such squiggly, long or thin shapes in nature, and this is often an adaptation to get around the surface-to-volume decrease issue. Examples include your intestines, or the interface between the placenta and the uterus.