Imagine $a \times b$ was not defined and we need to come up with something.
Here is the justification I could come up with:
Suppose, I have to measure this thing called area, I have to come with a notion of a unit area just as there is a unit value for a number. The reason is that if we have to measure something, there has to be some fundamental building block and we express everything in terms of. So the easiest way to define a unit area is to take a 1x1 rectangle. Now, if I just represent the number $a$ on the x-axis of the Cartesian coordinate system, I still don't have an area because it has only one dimension. So I just go to 1 on the y-axis and draw a line so that I can come up with something! Now I can see that there are $a$ unit squares in there. Extending this to $b$, I would have $a \times b$ unit areas. Is that good way to think of why the area is defined as $a \times b$?
I think the key was to idea build this fundamental notion of a unit area!
I would like to know if there is a better way to convince that $a \times b$ makes the most sense to measure area!