Why $f(x) = \sqrt{x}$ is a function? Why $f(x) = \sqrt{x}$ is a function (as I found in my textbook) since for example the square root of $25$ has two different outputs ($-5,5$) and a function is defined as "A function from A to B is a rule of corre-
spondence that assigns to each element in set A exactly one element in B.", so $f(x) = \sqrt{x}$ is not a function?

I will assume that in this portion of your textbook it is assumed that $x \in \mathbb{R}$, and with that condition $f(x)=\sqrt{x}$ is certainly a function. Specifically $f:[0,\infty) \rightarrow [0,\infty)$. It meets the formal definition of a function (not one to many).
Your confusion is due to an inappropriate extrapolation of reasoning. Specifically this:
You know that $x^2=25 \Rightarrow x= \pm \sqrt{25}$ by the square root property. However this function in no way involves taking the square root of both sides of an equality. It is just a function $f(x)=\sqrt{x}$, and the domain is $x \geq 0$ by virtue of the fact that you are living in the real number system in this portion of your textbook.
Now $f(x)= \pm \sqrt{x}$ is certainly not a function. For example, if the question were "Let $y^2=x$. Is $y$ a function of $x$?" You would say no in this case as $y= \pm \sqrt{x}$, and you would choose $x=25$ to counter the definition of a function.
Note, your statement "the square root of $25$ has two different outputs" is false. There is only one output. However, if $y^2=25$, then $y$ has two different solutions.   
