First let's do injectivity (which will be easy, and so correspondingly not so interesting). Since $K$ exhausted by finite type $k$-subalgebras $K_i$ (definitely not fields in general), we have $X(K)= \varinjlim X(K_i)$ and $G(K)= \varinjlim G(K_i)$ (as $X$ and $G$ are finite type, or alternatively it is clear in the affine case). Thus, $X(K)/G(K) = \varinjlim X(K_i)/G(K_i)$, so it enough to treat the $K_i$ in place of $K$. So we can assume $K$ is finitely generated as a $k$-algebra. [This is a powerful idea, even when the original $K$ is a field.] By the Nullstellensatz there is a $k$-algebra map $s:K \rightarrow k$ (quotient by any maximal ideal) with $k \rightarrow K$ as section; this is the "specialization" trick. It defines a map of sets $X(K)/G(K) \rightarrow X(k)/G(k)$ with the original map $j$ as a section (as $A \rightsquigarrow X(A)/G(A)$ is a functor on $k$-algebras $A$), so $j$ is injective.
Say $x_1,\dots,x_n$ in $X(K)$ lie in distinct orbits. Exhausting $K$ by finitely generated $k$-subalgebras $K_i$ as above, we can find a big enough $K_i$, call it $A$, so that $x_1,\dots,x_n \in X(A)$. We want to show that for a "sufficiently generic" specialization map $A \rightarrow k$, their images in $X(k)/G(k)$ remains distinct. Here, the valuable geometric intuition (which makes sense even within classical algebraic geometry, since $k$ is algebraically closed and $S :=$ Spec($A$) is basically a classical irreducible variety, as $A$ is a domain of finite type over $k$) is that the $x_i \in X(A)$ are sections to the projection $X \times S \rightarrow S$ such that on the geometric generic fiber over $S$ (i.e., pullbacks along $A \rightarrow K$) they are isin pairwise distinct $G$-orbits, and we want to claim that under specialization over some dense open in $S$ they remain in pairwise distinct $G$-orbits. In other words, we aim to "verify" an instance of the
Principle of the Geometric Generic Fiber: for a finite collection of finite type schemes over an irreducible noetherian scheme $S$, and any "finite information" structure involving them (maps among them, coherent sheaves on them, etc.), any reasonable property of this structure that holds over a geometric generic point of $S$ also holds on fibers over the geometric fibers supported in some dense open in $S$.
[In practice it isn't always obvious that certain properties are "finite information", such as flatness or surjectivity of $S$-maps, but EGA IV$_3$ lays out the whole story on this principle.]
Now you can see the one serious ingredient that uses schemes (going beyond classical algebraic geometry) in an essential way: the validity of Chevalley's theorem on constructible images in the scheme framework, and the ability to apply it in conjunction with the literal generic point (and geometric points over that). Hopefully you can see that (together with specialization) this is a broadly useful technique for propogating results from an algebraically closed extension of an algebraically closed field back down to the ground field (such as surjectivity on points valued in an algebraically closed field). And that once one realizes this idea, it is sort of simple in the end. In effect, the Principle of the Geometric Generic Fiber above (which is made precise in EGA IV$_3$) is the scheme-theoretic replacement for Weil's "universal domain" concept.
