Why a function from $\mathbb{R}^2$ to $\mathbb{R}$ is a surface while from $\mathbb{R}$ to $\mathbb{R}^2$ is a curve? This is a silly question, but I just can not see the reason.

The function from $\Bbb R$ to $\Bbb R^2$ has to be a curve (a unidimensional object) because you are encoding information about a single variable. So, even a function from $\Bbb R$ to $\Bbb R^3$ will also be a curve in 3-D. 
However, the function $\Bbb R^2$ to $\Bbb R$ encodes information from two variables into one. Hence it is represented as a surface (a 2-dimensional object).
