In Milne's EC, he defines an etale neighbourhood of a point $x \in X$ by a pair $(Y,y)$ with an etale morphism $f: Y \to X$ where $f(y) = x$, such that $k(x) = k(y)$.

What I don't understand is this last condition. Is there an intuitive reason why this condition is necessary? Explanations are welcome, although examples would be more appreciated.