The short answer is the following: for a $G$-torsor $E_G$ over $C$ and for the associated projective scheme $E_{G,B} := E_G/B$, a lift $E_B$ of $E_G$ to a $B$-torsor over $C$ is the same thing as a section $\sigma:C\to E_{G,B}$ of the projection $\pi:E_{G,B}\to C$. Via infinitesimal deformation theory of the Hilbert scheme, this section is unobstructed if $H^1(C,\sigma^*(\Omega_\pi)^\vee)$ is zero. Finally, $\sigma^*(\Omega_\pi)^\vee$ turns out to be $E_B \times^B \text{Lie}(U^{-})$. In fact, since $E_B$ has a further reduction of structure group to a maximal torus $T$, i.e., $E_B$ equals $B \times^T E_T$ for a $T$-torsor $E_T$, the bundle $E_B\times^B \text{Lie}(U^{-})$ turns out to equal $E_T\times^T \text{Lie}(U^{-})$, which splits as a direct sum of invertible sheaves on $C$ (because every representation of $T$ is a direct sum of characters). Thus $E^T\times^T \text{Lie}(U^{-})$ has vanishing $h^1$ if and only if each of these summands has vanishing $h^1$. That is precisely the condition imposed by Gaitsgory and Nadler to define the open subset $\text{Bun}_{B,r}$ inside $\text{Bun}_B$.