A rectangle comprises two horizontal lines and two vertical lines. Count the number of pairs of horizontal and number of pairs of vertical lines possible in a chess board to get the answer.
There are $9$ vertical lines and $9$ horizontal lines on the chess board.
To form a rectangle you must choose $2$ of the $9$ vertical lines and $2$ of the $9$ horizontal lines.
For the two horizontal lines: the first line can be chosen in $9$ ways the second in eight ways. This would imply that you could tell the difference between lines $1$ and $3$ say and $3$ and $1,$ which is not the case so you need to divide $9 \times 8$ by $2,$ making $36.$
Similarly you can choose the two vertical lines in $36$ ways.
So the number of rectangles is given by $36 \times 36$
$\Rightarrow 36 \times 36 = 1296$ rectangles.