$RP^n=S^n/{\sim}$ where $x\sim-x$. Let $p:S^n\rightarrow RP^n$ and this map is locally trivial with the fibre the two point set.
Say if $n=2$, then $RP^{2}$ will be lines passing through the origin in $R^3$ and $S^2$ is the sphere.
If this map is locally trivial, then it mean that for an open neighbourhood $U$ in the base space, the preimage $p^{-1}(U)$ behaves like $U\times F$ where $F$ is the fibre.
Visually what is this $U$ like? Is it like a "bundle" of lines through origin? Like two cones with their vertices touching each other? Also, how to visualize the fibre $F$ and why is the preimage (a region of the sphere I assume) like $U\times F$? What is this "two point set"?
Appreciate if anyone can help in the visualisation. Thanks!