Solution:

All the rectangles on the board can be identified by connecting:
2 points of the 8 in the top edge (to form the length of the rectangle) and
2 points of the 8 in the left edge (to form the breadth of the rectangle).

To gain a better understanding, consider the 8 × 8 chess board (see animation above).
Note that there are 4 possibilities for the lengths of the rectangles to be 5 units.

The following table shows the number of possibilities for different lengths of the rectangles on a 7 × 7 board:

Length of rectangle	Number of Possibilities
7 units	1
6 units	2
5 units	3
...	...
1 unit	7

So, number of possibilities for different lengths of rectangles = 1 + 2 + 3 + ... + 7 = 28.
Similarly, number of possibilities for different breadths of rectangles = 1 + 2 + 3 + ... + 7 = 28.
Hence, number of rectangles = 28 × 28 = 784.

Food for thought:

Is there a formula for the sum of the first n positive integers ?
Is 1 + 2 + 3 + 4 + ... + n = n (n + 1) / 2 ?

Can this puzzle be solved quickly with knowledge of permutations and combinations?
Note ⁿC₂ is the number of combinations of n things taken 2 at a time.
ⁿC₂ = n (n − 1)/2. Hence the number of rectangles = ⁸C₂ × ⁸C₂ = 28 × 28 = 784.

Can you figure out the following alternative formula to solve this puzzle?
Number of rectangles on n × n board
= 2 (Sum of the products of all pairs of numbers from 1 to n) − (Number of squares on the board)

So, how many squares on a n × n board? Click here to find out.