Since $\mathbb{S}^3$ is conformally flat, we can think that any point of our manifold $M$ admits a neighborhood that is conformally equivalent to an open set in $\mathbb{S}^3$.
If two such neighbohoods overlap then the corresponding gluing map between corresponding open sets in $\mathbb{S}^3$ is a Moebius transformation.
So after applying a Moebius transformation to one of the open sets you can assume that the gluing map is identity.
This way you can extend the parametrization of a neighborhood to an immesed neighborhood of any path.
Since $M$ is simply connected it gives a well defined conformal immersion $M\hookrightarrow \mathbb{S}^3$.