On Wikipedia, I read, "A box can be thought of 'small boxes' infinitely repeating in all three dimensional directions"
I don't understand what does Wikipedia wants to say with a box containing infinite numbers of similar small boxes. Why would it be infinite. I can have any integer however.
Can any one explain it to me?
It is the least volume consuming and repeating structure of particles of any solid. So it's not the least volume out of everything, it is just the substance with the least volume with meets that criteria. The reason for this is because cells need a high surface area to volume ratio so that they can easily transport waste across the cell and dispose of it, as well as so that they can quickly utilize the nutrients coming in. Imagine the cells are a cube with side lengths $x$, then the surface area to volume ratio is $$6x^2:x^3$$ And now you can see why smaller values of $x$ benefit the cell. Because for $x<0$, we know that $x^3<x^2$, which means it would have a higher surface area to volume ratio.
I think the Wikipedia definition is more math centered, as that fundamental thinking process is essentially what Riemann integrals come down to in Calculus. However for understanding your definition I think you were forgetting the second part; they arent the least volume thing, but rather the least volume thing that is a consuming and repeating structure of particles.
Another Note
Another quick thing to note, is why cells are circular (approximately) rather than cubes. If we have a circle of radius $r$ and a cube of side length $x$ and we have their volume to surface area ratios $$4\pi r^2:\frac{4}{3}\pi r^3$$ $$6x^2:x^3$$ And lets at the case when the two volumes are the same so that we can see which gets better surface area. When they are the same we have $$\frac{4}{3}\pi r^3=x^3$$ Therefore $$r=x\left(\frac{3}{4\pi}\right)^{1/3}$$ Then we see that when the volumes of two figures are their same, the cubes surface area to the spheres surface area ratio is $$4\pi\left(\frac{3}{4\pi}\right)^{2/3}x^2:6x^2$$ So clearly spheres get better surface area to volume ratios.