By dual pixel they mean a horizontal line that is one pixel tall and two pixel wide. Each cell's side is 7 such dual pixels arranged diagonally. But since you stack tiles so that the grid line overlaps, it means that two 7-wide cells are together only 13-wide (and not 14). The top dual pixel of one is the bottom dual pixel of the other.

Going smaller, two 4-wide cells will again lose one line to the overlap and thus join to make a 7 (that's how 7 divides by 2).

Three 3-wide cells join and lose two lines to overlap and thus again join to make 33-2 = 7

Six 2-wide cells join and lose fives lines to overlap 62-5 = 7.

Or in general for n cells, there are n-1 overlaps, so if they are k-wide the combined cell will be n*k - (n-1) wide.

It makes more intuitive sense when you notice 7-1 = 6.

All "outer" dimensions are 6*n + 1.

So 4 cells are actually 4*6+1 not 4*7

So you can see that 4 cells can be divided into 3 parts in this way:

3*4*6+1 = 3*4*2*3+1 = 3*2*4*3+1

All of that is because there's 1px overlap between neighboring cells.

In spite of what the page says, the cell isn't 7x7. One diamond-shaped cell is 24 pixels wide, measuring from the center of the leftmost pixel to the center of the rightmost pixel, and 12 pixels tall center-to-center.

You mean that a cell has a diameter of 24 pixels. “Width” would be measured along one of the (angled) sides of the cell.