The idea is that different lifts of the function $f$ are determined by the choice of one point.
For example, consider the standard covering of $S^1$ by $\mathbb{R}$ (see page 29 in the text). Let $\gamma:[0, 1]\to S^1$ be the path traversing the circle once in the counter-clockwise direction. Then $\gamma$ can be lifted to any interval $[n, n+1]$ on the real line. Suppose $\gamma_2$ starts at $1$ and spirals up to $2$, and $\gamma_1$ starts at $0$ and spirals down to $1$.
The statement says that if the two lifts do not agree at a point, then the neighborhoods that are mapped homeomorphically to $S^1$ are disjoint.
Think about the pre-images of $-1$ under the covering map, corresponding to $\gamma(\frac{1}{2})$. Then $\gamma_2(\frac{1}{2})=\frac{3}{2}$, while $\gamma_1(\frac{1}{2})=\frac{1}{2}$. In particular, the neighborhoods of these points that map homeomorphically onto the corresponding neighborhood of $\gamma(\frac{1}{2})=-1$ are disjoint.