Let $f,g:\mathbb{R}^n\to \mathbb{R}$. What is $fg$? Is it function $\mathbb{R}^n\to \mathbb{R}$ or $\mathbb{R}^{2n}\to \mathbb{R}$?
For example, taking $n=2$ let $f(x_1,x_2)=x_1+x_2+x_1x_2$ and $g(y_1,y_2)=y_1+y_2-y_1^2 y_2^3$ we see that $fg:\mathbb{R}^4\to \mathbb{R}$.
Can anyone explain this confusing moment to me in detail?
If $f : \mathbb{R}^n \to \mathbb{R}$ and $g : \mathbb{R}^n \to \mathbb{R}$, then $fg : \mathbb{R}^n \to \mathbb{R}$ given by $(fg)(x) = f(x)g(x)$. It is key here that the codomains of $f$ and $g$ are $\mathbb{R}$ where it makes sense to multiply elements.
In your example, $fg : \mathbb{R}^2 \to \mathbb{R}$ is given by
$$(fg)(x_1, x_2) = f(x_1, x_2)g(x_1, x_2) = (x_1 + x_2 + x_1x_2)(x_1 + x_2 - x_1^2x_2^3).$$