This is exercise 11.4.C of Ravi Vail's note
Let $\pi :X\to Y$ be a surjective closed map of varieties with $Y$ irreducible and the fibre of $\pi$ are irreducible of the same dimension, then $X$ is irreducible.
and a solution can be found in this post Use irreducible fibers to show $X$ is irreducible.
I would want to ask is there any intuitive reasoning why this is true? Is this true for topological spaces in general?
Thanks in advance.