Question: I don't understand why this is true:
If $f:[0,1]\rightarrow[0,2]$ is a continuous function then exists $x \in [0,1]$ such that $f(x)=2x$

I don't understand why such a point exist. Why is there not for example a fixpoint $f(x)=x$?

Answer: 
Why is there not a fixed point $f(x)=x$? Nobody said there wasn't.

To show there does exist an $x$ with $f(x)=2x$, let $g(x)=f(x)-2x$. Then $g(0)=f(0)\ge 0$, while $g(1)=f(1)-2\le0$. So the intermediate value theorem shows there exists $x$ with $g(x)=0$.

(Or, if you have a theorem saying any map from $[0,1]$ to itself has a fixed point, consider $f/2$.)
There could be both!

The existence of such a point is a consequence of the intermediate value theorem.