Intuition behind definition of K-K asymptotic flatness I am reading some notes on black holes, and am confused by this definition of Kaluza-Klein asymptotic flatness:

If a spacetime $(M, \mathbf{g})$ contains a spacelike hypersurface $\mathscr{I}_{ext}$, diffeomorphic to $\left(\mathbb{R}^N\backslash\bar{B}(R)\right)\times Q$, where $\bar{B}(R)$ is a closed coordinate ball of radius $R$ and $Q$ is a compact manifold, we say that $\mathscr{I}_{ext}$ is a Kaluza-Klein asymptotic end. If a spacetime contains such an end, let $\tilde{h}$ be a fixed Riemannian metric on $Q$, and let $\tilde{g}=\delta\,\oplus\,\tilde{h}$, where $\delta$ is the Euclidean metric on $\mathbb{R}^N$. The spacetime is then said to be "Kaluza-Klein asymptotically flat" if for some $\alpha>0$, the metric $\gamma$ induced by $\mathbf{g}$ on $\mathscr{I}_{ext}$ and the extrinsic curvature tensor $K_{ij}$ of $\mathscr{I}_{ext}$, satisfy the fall-off conditions:
$\gamma_{ij}-\tilde{g}_{ij} = O_d(r^{-\alpha-\ell})\,,\;K_{ij} = O_{d-1}(r^{-1-\alpha-\ell})$
where $r$ is the radius in $\mathbb{R}^N$ and we write $f=O_d(r^\alpha)$ if $f$ satisfies $\tilde{D}_{i_1}...\tilde{D}_{i_\ell}\,f = O(r^{\alpha-\ell})\,,\; 0\leq\ell\leq d$, with $\tilde{D}$ as the Levi-civita connection of $\tilde{g}$.

In this context, what exactly is the significance/physical intuition of $Q$, and why do we take $\bar{B}(R)$ rather than $B(R)$ as in the definition of normal asymptotic flatness?
 A: The physical intuition of $Q$ comes from Kaluza-Klein reduction, which reduces a higher dimensional field theory to a lower dimensional one. For a complete reference, you may want to look here.
In short, our world seems to have $4$ dimensions, but if we start with a theory that requires higher dimensions, we might wonder how that reconciles with our daily perception. Kaluza-Klein reduction gives one way—it proposes the ground state of the metric, which satisfies the classical field equations, to be of the following form (there can be wrapped factors sometimes):
$$ds^2=ds^2_{M_4}+ds^2_{M_{d-4}},$$
with the ground state of the spacetime to be $M_4\times M_{d-4}$, with $M_{d-4}$ compact, so that its effects are controlled by its characteristic scale which can be related to the Planck length, i.e. it is too small to be actually seen in our daily life. That is usually what the "curled extra dimensions" means. In the definition you gave, $Q$ is more or less the "curled extra dimension."
For the $B(R)$ and $\bar{B}(R)$ problem, it seems to be more or less a choice of definition: in both ways, we can extend $R$ a little bit to obtain the other.
