Why random variables is a function? It seems that it violates the definition of function.

It is not a violation of the definition of a function when two unequal inputs return equal outputs. Rather, the rule is that a single input cannot return more than one output!

For example, when one first is introduced to functions, one is mostly paying attention to things like $f(x) =x^2$, which defines a function from the set of real numbers to the set of real numbers. In this case, there are two "students", say the inputs $x=1$ and $x=-1$, which have the same "height", both inputs give the output of $1$. This is not a problem and $f(x)=x^2$ is definitely a function. The same is true in the context of random variables. Your reading of the definition is just slightly off, is all.

Function is Random variable when pre-image of it's any value is measurable set i.e. probabilistic events. So we can say that function takes it's value with some probability.

Formally lets take $(Omega, mathcal{B}, P)$ probability space and random variable $X:Omega to mathbb{R}$. Then $P(X^{-1}(a))$ is probability with $X$ is obtain value $a$.

So random variable of course have fixed rule for input-output connection, it's usual function, but on another hand, defined on probability space it characterize the random nature of measuring.