The following two are definitions of continuity of a function:
1) The function is continuous at every point $c$ in the domain. The function is continuous at a point $c$ if for any given $\epsilon > 0$, we can find a $\delta$ such that $\forall x \in (c-\delta, c+\delta)$ $ |f(x) - f(c)| < \delta$.
2) Inverse image of every open set is open.
Now, I have gone through the proof that these two are equivalent from the Rudin's book, but, how to understand this equivalence word by word intuitively. Which words in $(1)$ correspond to which word in $(2)$.
First, notice that any open set in $\mathbb{R}$ is a union of open intervals. Thus, we will only consider intervals. Given an $\epsilon$ and $\delta$, you've got that $(a-\delta,a+\delta)$ and $\left(f(a)-\epsilon,f(a)+\epsilon\right)$ are open intervals containing $a$ and $f(a)$. Your definition $(1)$ then can be seen to be equivalent to the fact that given an open interval around $f(a)$, there is an open interval around $a$, call it $I$, small enough so that $f(I)\subseteq (f(a)-\epsilon,f(a)+\epsilon)$. This $I$ is your $(a-\delta,a+\delta)$. But, this says that $f^{-1}\left(f(a)-\epsilon,f(a)+\epsilon\right)$ is a union of open intervals, hence open.