Confusion about geometric interpretation of proof that $\mathbb R[X,Y,Z]/ \left\langle X^2+Y^2+Z^2 -1 \right\rangle $ is a UFD I'm working through a proof that $R=\mathbb R[X,Y,Z]/ \left\langle X^2+Y^2+Z^2 -1 \right\rangle $ is a UFD. The idea is to localize at $1-x$ and show the result is a UFD. Since $R$ is atomic as a quotient of a Noetherian ring, and $1-x$ is prime, Nagata's lemma will imply the resut. 
Let $x,y,z$ be the images of $X,Y,Z$ in the quotient.
By exactness of localization $R_{1-x}\cong \mathbb R[X,Y,Z]_{1-X}/ \left\langle X^2+Y^2+Z^2 -1 \right\rangle$. If I understand correctly, $\mathbb R[X,Y,Z]_{1-X}$ means we're deleting the plane $X=1$ from $\mathbb R^3$.
Take $T=(1-X)^{-1}$ and note $\mathbb R[X,Y,Z]_{1-X}\cong \mathbb R[X,Y,Z,T]$. Descending to $R_{1-x}$ as the quotient above, some manipulations enable me to show $\mathbb R[x,y,z,t]\cong \mathbb R[ty,tz,t^{-1}]$, and since the latter is the localization at $t$ of a UFD, Nagata ends the proof.
I would like to understand what exactly is going on geometrically here and I really have no clue where to start because already at the quotient $R_{1-x}\cong \mathbb R[X,Y,Z]_{1-X}/ \left\langle X^2+Y^2+Z^2 -1 \right\rangle$ I have no idea what I should be visualizing.

Given a noetherian integrally closed domain $A$ we have the equivalence $$ A \operatorname { is a } UFD \iff Cl(A)=0$$where $Cl(A)=0$ is the class group of $A$.
If your case $\operatorname {Spec}(R)$ is smooth so that $Cl(R)=Pic(R)$, the Picard group of $R$.
So the problem boils down to proving that the algebraic Picard group $Pic(S^2)$ of a sphere is trivial.
I don't know a  way of showing that simpler than the calculation you made but intuitively this is not so surprizing since all continuous (or differentiable) line bundles on the sphere are trivial because $H^1(S^2,\mathbb Z/2\mathbb Z)=0$.
Contrastingly the ring $\mathcal O(S^1)=\mathbb R[X,Y]/\langle X^2+Y^2-1 \rangle$ is not a UFD since the circle $S^1$ possesses a non-trivial line-bundle, the notorious Möbius line bundle, which can be given an algebraic structure.
