[dagss]

Not sure if it helps, but trying: In a sense area isn't anything else than multiplication. It isn't like you have a) multiplication and b) area and then prove that a=b.

Rather, area IS multiplication.

The unit of "square meter" quite literally means "meter multiplied by meter".

[quietbritishjim]

I agree and I'd say that area is almost, in some intuitive way, the more basic thing and multiplication follows from that (although I know that's not mathematically true). The definition of multiplication for natural numbers is repeated addition (e.g. 3 x 5 is defined to be 5 + 5 + 5). Many people would see that as the count of a 3 by 5 grid of objects, and that's certainly how we'd explain the commutativity of multiplication in school. If those individual objects happen to be unit squares then you have area.

[umanwizard]

In fact, it wouldn’t surprise me if multiplication was first invented to calculate areas (e.g. of agricultural fields).

[Someone]

I would say area is integration, and that, for the simple case of integrating a constant function, equals multiplication of that constant by the length of the interval being integrated over (measure of the set, if you’re doing Lebesgue integration)