I don't understand why the surface element isn't only $\|\vec{r}_u \times \vec{r}_v||$ isn't instead of $\|\vec{r}_u \times \vec{r}_v|| \Delta u \Delta v$. The "||" denotes the vector norm. The "$ \times$ " is the cross product. $\|\vec{r}_u \times \vec{r}_v||$ is already the area of the parallelogram calculated by crossing unitary vectors. Why do we need to scale the area of the parallellogram by "$\Delta u \Delta v$"?
Why isn't the surface integral $\iint \|\vec{r}_u \times \vec{r}_v|| $? without "$ du dv$"?
The surface element in the plane is an infinitesimal rectangle $du\,dv$. The area of a plane figure $A\subset\Bbb R^2$ is $\int_Adu\,dv$.
The surface element of a surface $\vec r\colon A\to\Bbb R^3$ is the image under $\vec r$ of the infinitesimal area rectangle $du\,dv$. This image is a parallelogram with sides $\vec r_u\,du$ and $\vec r_v\,dv$; its area is then $|(\vec r_u\,du)\times(\vec r_v\,dv)|=\|\vec r_u\times\vec r_v\|\,du\,dv$, and the area of the surface $\int_A\|\vec r_u\times\vec r_v\|\,du\,dv$.