Why is $\text{meter} \times \text{meter}$ a legitimate unit for measuring area? How did we transition from meters, as a unit of distance, to $\text{m}^2$, as the unit of area, I do understand that for example if we define a square that is 1meter long and 1meter wide and use it as our basic unit for measuring areas, then the area of a rectangle (expressed as the number of my meter-squares)would simply be its length times its wide. But that aside, what justifies $\text{m}\times \text{m}$ as a unit for measuring the area, in other words, why a distance "times" another distance yields an area?

All the theorems are just mere consequences of axioms which have no proof and must be assumed(they are not silly). Now the area of a square of $1m×1m$ is **defined** to be $1m^2$. So now in any figure, you may fit all these squares to get the actual "area". So you actually multiply with numbers $>1$, which signifies you are fitting that many unit squares in your figure.