In what sense do Jones' original subfactors come from quantum SU(2)

You can construct a subfactor (under very mild assumptions) from an object $X$ in a rigid C*-tensor category, by taking the limit of inclusions $$ {\rm End}(X^{\otimes n}) \simeq \{ \iota_X \otimes T \mid T \in {\rm End}(X^{\otimes n})\} \subset {\rm End}(X^{\otimes n+1}) $$ as $n \to \infty$. One good entry point is the Longo-Roberts paper [LR97], I guess. In particular, you get Jones's subfactors from the "half spin" object in ${\rm Rep}({\rm SU}_q(2))$ for root of unity $q$.

[LR97] R. Longo and J. E. Roberts, A theory of dimension, K-Theory 11 (1997), no. 2, 103–159. MR 1444286 (98i:46065)

The fusion category $\mathcal{C}_\ell$ of unitary highest weight projective representations of level $\ell$ of the loop group $LSU(2)$ is equivalent to ${\rm Rep}({\rm SU}_q(2))$ with $q = e^{\frac{i \pi}{\ell + 2}}$ (see this paper, first paragraph p5).

Now, for any simple object $\rho$ of $\mathcal{C}_\ell$ (characterized by its spin $i \le \ell/2$), there is a Jones-Wassermann subfactor (see this Jones' survey Section 6, or also this answer): $$ \rho (L_I)'' \subseteq \rho (L_{I^c})'$$ of index $\frac{sin^{2}(p\pi/m)}{sin^{2}(\pi/m)}$ with $m=\ell + 2$ and $p=2i+1$.

At spin $1/2$, it is exactly the Jones's original subfactor of index $4cos^2(\frac{\pi}{m})$.