here's a run-down. consider the open subset $U_0=\{[x_0\colon\dotsc\colon x_n]\in\mathbb{P}^n\mid x_0\neq0\}$ (this is well-defined!). these are precisely the lines in $\mathbb{R}^{n+1}$ that pass through the hyperplane $\{x_0=1\}$ and each such line is uniquely determined by its one point of intersection with that hyperplane. check that this gives a homeomorphism $U_0\cong\{x_0=1\}\cong\mathbb{R}^n$. the complement of this open subset are precisely the lines parallel to that hyperplane, which are the lines in $\{x_0=0\}=\mathbb{R}^n$. check that this gives a homeomorphism $\mathbb{P}^n\set…

Ah no, what I said above is incomplete. So $f$ and $g$ are constructed from two coordinate patches $U$ and $V$. I wanna work on $U\cap V$ with the coordinates of $V$, so in these coordinates $g$ just is projection on the last coordinate, i.e. $g=x_n$ (using sub- instead of superscripts now).
then, the local expression reads $\frac{f(x)}{g(x)}=\frac{\partial f}{\partial x_n}(0)+\frac{R(x)}{x_n}$. it's actually still not clear this can be extended to $x_n=0$, though. Taylor doesn't give me a uniform estimate on the remainder, hmm.

Well so do I. From what I understand it is some funny business about how moduli spaces of stable maps might in general be too big/singular and have arbitrary components. So in order to get enumerative invariants out you want to integrate classes against fundamental classes on the moduli space of stable maps. This would require you to define a "correct" notion of fundamental class on this usually horrible space.

This is usually done by hoping for a fantasy. That is "locally" embedding your moduli space into some smooth projective variety, looking at some vector bundle such that some section…