I will use the elementary school definition of area as the number of unit squares you can fit into it.
We have a rectangle with dimensions a,b and we want to prove its area is a*b.
Let's first assume the rectangle has natural number dimensions a,b.
Assume the height b is 1. Then we simply stack up a squares horizontally to fully fill the rectangle.
Now if we increase b we just add one new such row, so the answer is a more. By induction, we have the area is A=ab.
Now assume the rectangle has rational number coordinates.
a=p/q, b=s/t, p,q coprime, s,t coprime.
We stack up q such rectangles horizontally and stack up resulting figure t times vertically, to get a rectangle of width p and height s. Its area is ps.
But each smaller rectangle which makes up the big rectangle is congruent and so has the same area. Since there are qt such rectangles, the resulting area of an individual rectangle is ps/qt = ab.
Since the real numbers are fake we won't bother with them.