The Gauss-Codazzi equations relate the intrinsic & extrinsic curvature of a submanifold $\Sigma$ to (the projection of) the curvature of the manifold in which it is embedded. Specifically, it can be shown from the Gauss-Codazzi equations that
$$
R = \bar{R} + (\bar{K}^2 - \bar{K}_{\mu \nu} \bar{K}^{\mu \nu}) + 2 \nabla_M\left( n^N \nabla_N n^M - n^M \nabla_N n^N\right) \tag{$\star$}
$$
where $\bar{R}$ is the intrinsic Ricci scalar on $\Sigma$, $\bar{K}_{\mu \nu}$ is the extrinsic curvature of $\Sigma$, $\bar{K} \equiv \bar{K}_\mu {}^\mu$, and $n^N$ is the unit normal of $\Sigma$. A full derivation of this result can be found in Chapter 3 of A Relativist's Toolkit by Eric Poisson (2004).
It is also possible to show from the metric (13) that
$$
\det (-G) = \det(-\bar{g}) e^{-8 k T |\phi|} T^2.
$$
(The factor of 8 comes from a factor of $e^{-2 k T |\phi|}$ for each brane dimension.) Thus, the 5-D Einstein-Hilbert action includes the term
$$
S = \int d^4x \int d\phi \sqrt{-G} (2 M^3 R) \supset \int d^4 x \int d\phi \sqrt{-\bar{g}} \left( 2 M^3 e^{-4 k T |\phi|} |T| \bar{R} \right)
$$
and if we assume that $T$ takes on its VEV of $r_c$, we obtain the low-energy effective gravitational action in (15).*
The extrinsic curvature terms in the starred equation above would (likely) manifest as some kind of Kaluza-Klein modes; I would expect that they are examined in detail in the subsequent paper that was "in preparation" at the time that this preprint was posted (reference [9] in the preprint.) The last term is, of course, a total derivative and does not affect the classical equations of motion.
*Almost. It seems to me that the 2 in the factor of $e^{-2 k r_c |\phi|}$ in (15) should be a 4 instead (or, more generally, the dimension of the submanifold.) But it's possible that there's an explanation for it that I'm not seeing right now. In any event, it does not make a major difference for their subsequent hierarchy arguments.
