Here is an extremely useful fact: if $X$ is a regular variety and $f \colon X \dashrightarrow \mathbf{P}^n$ is a rational map defined on a dense open $U \subset X$, then $\mathcal{f}$ extends uniquely to a maximal open subset $U'$ of $X$ and the complement of $U'$ has codimension at least two. In particular, when $X$ is a smooth curve there is a unique extension to the whole curve. Now apply this to the rational map from $X$ to the complete curve associated to $k(X)^G$, which you know can be embedded in a projective space. I don't have the book on me but this is certainly somewhere in Hartshorne. The proof goes by using that the local ring of a point in codimension one is a DVR, so you can write down an explicit extension of the map $f$ in codim 1 by multiplying through by the lowest possible power of a uniformizer.
On a sidenote, this useful fact is already needed to show that there is a unique smooth complete curve associated to a function field in one variable.
Edit. I think I see now what you are getting at: you want to know why the points of $X/G$ correspond to $G$-orbits on $X$, right? (Knowing this, it immediately follows that if $P$ is a ramification point of order $e_P$, then it has $|G|/e_P$ conjugates under $G$.) There is a general theorem of Emmy Noether on quotients of affine varieties by finite groups which will show this, which can be found in any book on commutative algebra. But for curves I guess you could make a direct proof as follows: pick points $P$ and $Q$ on $X$ in different orbits. Since the function field separates points you can choose a function that vanishes at $P$ but not at $Q$. Do the same thing also for the conjugates of $Q$. Then a linear combination $F$ of these functions is nonzero on all conjugates of $Q$. Take the product of all conjugates of $F$ under the $G$-action: this is a $G$-invariant function which vanishes on the orbit of $P$ but not on the orbit of $Q$. The result follows.
I am surprised that nobody has mentioned (yet) that an essential point of the algebro-geometric notions of (algebraic) stack and algebraic space is to completely shift the burden of construction problems: one gives up on trying to make any kind of actual ringed space at all, and in fact the whole point is to create a kind of "geometry for functors". In particular, Bun$_G$ is demoted to the status of a definition (there's no "space" to be constructed: one just defines a certain groupoid-valued functor) and the real content is to show that it is "near enough" to representable functors that one can import to it many concepts from algebraic geometry.
It is easiest to explain this with an example. Let's contrast the approaches to the "Hilbert scheme" for a projective scheme $X$ over a field $k$ (to fix ideas). In the original approach of Grothendieck, he defined a functor $\underline{\rm{Hilb}}_{X/k}$ on the category $k$-schemes, namely
$\underline{\rm{Hilb}}_{X/k}(S)$ is the set of finitely presented closed subschemes $Y \subset X \times S$ such that the structure map $Y \rightarrow S$ is flat; this is a contravariant functor of $S$ via pullback (i.e., for any $f:S' \rightarrow S$ over $k$, the map
$$\underline{\rm{Hilb}}_{X/k}(f): \underline{\rm{Hilb}}_{X/k}(S) \rightarrow \underline{\rm{Hilb}}_{X/k}(S')$$ carries $Y \subset X \times S$ to $Y \times_S S' \subset (X \times S) \times_S S' = X \times S'$).
The real substance in this approach is to show $\underline{\rm{Hilb}}_{X/k}$ is a representable functor. This amounts to constructing a "universal structure": a distinguished $k$-scheme $H$ equipped with an $H$-flat finitely presented closed subscheme $Z \subset X \times H$ such that for any $k$-scheme $S$ and any $Y \in \underline{\rm{Hilb}}_{X/k}(S)$ whatsoever there exists a unique $k$-map $g:S \rightarrow H$ for which the $S$-flat closed subscheme
$$Z \times_{H,g} S \subset (X \times H) \times_{H, g} S = X \times S$$
coincides with $Y$. In effect, $k$-maps to $H$ "classify" the functor $\underline{\rm{Hilb}}_{X/k}$.
But Grothendieck's representability result was proved via methods of projective geometry, and there were strong indications that one should seek a result like this more widely for proper $k$-schemes (beyond the projective case) yet nobody could find such a way to do this in the framework of schemes. Artin brilliantly solved the problem by widening the scope of algebraic geometry via his introduction of algebraic spaces, but it must be appreciated that in so doing one gives up on constructing a "representing space" in any sense akin to what Grothendieck does. In particular, algebraic spaces are not ringed spaces at all. Instead, they are Set-valued functors which are "near enough" to representable functors that it becomes possible to make sense of many concepts from algebraic geometry for these structures (and there is an associated topological space permitting one to speak of irreducibility, connectedness, etc.).
The crucial point is that in the Artin approach, the functor is the algebraic space. That is, in Artin's approach it is a complete misnomer to speak of "constructing the moduli space" for the Hilbert functor $\underline{\rm{Hilb}}_{X/k}$ for a proper $k$-scheme: the functor one has defined above literally is to be the algebraic space, so there is nothing to be constructed there. Instead, the real work is to build the representable functors "near enough" to this so that one can conclude that the functor one has already defined is indeed an algebraic space (and hence one can do meaningful geometry with it, after getting acclimated to the new setting).
Artin's breakthrough was to identify checkable criteria with which one could show (entailing some real work) that many functors of interest are "near enough" to representable functors to be algebraic spaces (and his PhD student Donald Knutson devoted his thesis to working out very many concepts of algebraic geometry in the wider setting of algebraic spaces, often entailing new kinds of proofs from what had been done for schemes). But even with $\underline{\rm{Hilb}}_{X/k}$ for projective $X$ there is a fundamental difference in the nature of the results: Grothendieck built the Hilbert scheme ${\rm{Hilb}}_{X/k}$ as a countably infinite disjoint union of quasi-projective schemes, so one knows the connected components are quasi-compact, whereas the Artin approach gives no information whatsoever on quasi-compactness properties for the algebraic space $\underline{\rm{Hilb}}_{X/k}$ (but of course it has the huge merit of being applicable far more widely, allowing to "do algebraic geometry" with many many more functors of interest; nonetheless, Grotendieck's quasi-compactness results for Hilbert schemes remain vital as a tool for proving quasi-compactness features of rather abstract algebraic spaces).
As another illustration, consider the functor $M_{g,n}$ of $n$-pointed smooth genus-$g$ curves. In the Mumford approach via GIT (say with $n$ big enough to get rid of non-trivial automorphisms), there is tremendous effort done to build a universal structure. In the Artin approach via stacks (which allows $n$ to be quite small too), the moduli problem is the stack: it is incorrect to speak of "constructing the moduli space" in this approach, since one has literally defined $M_{g,n}$ and there's all there is to it (granting that one is sufficiently fluent with descent theory and coherent duality to see at a glance that $M_{g,n}$ enjoys nice descent properties). But instead the real effort is to build appropriate "scheme charts" over $M_{g,n}$ to give precise meaning to a sense in which $M_{g.n}$ is "near enough" to representable functors that we can make sense of many concepts of algebraic geometry for this functor or for its groupoid-valued version when $n$ is small.
The situation with Bun$_G$ is similar: the groupoid assignment is the geometric object. Techniques of Artin provide a precise sense in which some schemes can be regarded as "smooth" over this gadget, with those used to define geometric concepts for Bun$_G$ in the same spirit as one uses open balls to define concepts for manifolds, but it is rather a misnomer to speak of "constructing Bun$_G$"! That is, one has to clearly distinguish the serious task of describing some "charts" on Bun$_G$ (with which to make computations and prove theorems) from the much more mundane matter of simply defining what Bun$_G$ is: it is the assignment to any $X$ of the groupoid of $G$-bundles on $X$, and the descent-like properties are easy to verify (so it is a "stack in groupoids"), and there is nothing more to do as far as "making the moduli space" is concerned. Of course, one cannot do anything serious with this until having carried out the real effort with Artin's criteria to affirm that this stack admits enough "scheme charts" to be an Artin stack and hence admit the rich array of meaningful concepts for it as in algebraic geometry (irreducible components, coherent sheaf theory, cohomology, smoothness properties, dimension, etc.) There is no gluing to be done: we define the global moduli problem at the outset. We have to do work to show this admits meaningful geometric concepts, and in practice there can be especially useful "scheme charts" or techniques or "local description" with which one can explore it. However, one really should not regard these explicit descriptions as "constructing Bun$_G$"; the construction is just the initial basic definition mentioned above. Another wrinkle is that in the algebro-geometric setting one has to use some serious Grothendieck topologies to make this all work, so it isn't as simple as with making manifolds by gluing open balls (likewise for the notion of $G$-bundle).
Of course, there are interesting and instructive analogies with constructions in homotopy theory, but to make coherent sense of actual algebro-geometric proofs involving Bun$_G$ one should recognize that this "space" is demoted to the status of a definition (in complete contrast with the setting for Hilbert schemes by Grothendieck, where he had to really build a representing scheme before geometry could be done) and the substantial effort is to build many kinds of interesting "scheme charts" over it with which to explore the geometric features of this stack. The framework of Artin stacks gives a systematic way to make sense of this process for many kinds of moduli problems encountered in practice. But at the end of the day stacks and their cousins are not ringed spaces (even though their structure can be explored using many auxiliary ringed spaces); e.g., even though there is an associated topological space with which to define various topological notions, in no sense is a map determined in terms of some kind of map of ringed spaces resting on those associated topological spaces (in contrast with the cast of schemes). The way one gets around the loss of contact with a specific ringed space is by a huge amount of descent theory. I think it is very important to always keep that in mind or else a lot of relevant issues in definitions and proofs will be rather confusing.
Best Answer
Here is a sketch of an argument which directly uses simple connectedness of $\mathbb P^1$, and is related to the simple connectedness of rationally connected smooth varieties mentioned by Sandor in one of his answers.
The idea is to treat the $\mathbb P^1$s in $\mathbb P^n$ as analogous to arcs in a topological space, and to make a lifting argument (just as one does in the basic topological theory of covering spaces).
Let $Y \to \mathbb P^n$ be a finite etale map. Fix a base points $x \in \mathbb P^n$ and a point $y \in Y$ lying over $X$. If $x' \in \mathbb P^n \setminus \{x\}$, there is a unique line $L$ joining $x$ and $x'$. The preimage of $L$ is a disjoint union of curves $L'$, each mapping isomorphically to $L$ (by simple connectedness of $\mathbb P^1$), and we can choose a unique $L'$ containing $y$. Now let $y'$ be the point of $L'$ lying over $x'$.
The map $x' \mapsto y'$ (and of course mapping our original point $x$ to $y$) gives a section to the given map $Y\to \mathbb P^n$, which is what we wanted.
Added: Here is one explanation of why the map $x' \mapsto y'$ is algebraic. Let $\pi:Y \to \mathbb P^n$ be our given etale map. First note that $x' \mapsto \pi^{-1}(L)$ (where $L$ is the line joining $x$ and $x'$, as above) is a morphism from $\mathbb P^n \setminus \{x\}$ to the Hilbert scheme of $Y$. Now picking out the connected component $L'$ of $\pi^{-1}(x')$ containing $y$ is a morphism from our given locally closed subset of the Hilbert scheme to the Hilbert scheme, and so altogether we see that $x' \mapsto L'$ is a morphism. Finally, mapping $L'$ to $x'$ (which can be described as forming the intersection $L' \cap \pi^{-1}(x')$) is again a morphism. So altogether we have a section $\mathbb P^n \setminus\{x\} \to Y$. One way to show that this extends as a section over all of $\mathbb P^n$ (by sending $x$ to $y$) is just to repeat the whole process for a different choice of $x$, and glue the two resulting sections.