This follows because the $d$-dimensional (global) conformal group ${\rm Conf}(d)$ is locally isomorphic to the proper Lorentz group $G:=SO(d\!+\!1,1)$ in Minkowski space $$\mathbb{R}^{d+1,1}~\cong~\mathbb{R}^d\times \mathbb{R}^{1,1}, \qquad \mathbb{R}^{1,1} ~\cong \underbrace{\mathbb{R}}_{\ni x^+}\times \underbrace{\mathbb{R}}_{\ni x^-},\tag{1} $$
cf. e.g. this Phys.SE post.
We can clearly embed the product subgroup
$$\begin{align}H~:=~SO(d)\times SO(1,1)~\cong~&\begin{pmatrix} SO(d) \cr & SO(1,1) \end{pmatrix}_{(d+2)\times (d+2)}\cr \subseteq ~&SO(d\!+\!1,1)~=:~G.\end{align}\tag{2}$$
Here the proper Lorentz group in 1+1D becomes diagonal
$$SO(1,1)~=~\left\{\begin{pmatrix} b \cr & b^{-1} \end{pmatrix} \in{\rm Mat}_{2\times 2}(\mathbb{R}) ~\mid~ b\in\mathbb{R}\backslash\{0\}\right\}~\cong~\mathbb{R}\backslash\{0\}\tag{3}$$
if we use light-cone (LC) coordinates $x^{\pm}$. The LC coordinates $x^{\pm}$ have weights $\pm 1$ because we identify boosts with conformal dilations. The sought-for eq. (A.7) is just the following branching rule
$$ \underline{\bf d+2}~\cong~ \underline{\bf d}_0\oplus \underline{\bf 1}_{1}\oplus \underline{\bf 1}_{-1} \tag{4}$$
of Minkowski space (1). The single box $\Box$ on the LHS and the RHS of eq. (A.7) denotes the defining representation of $SO(d\!+\!1,1)$ and $SO(d)$, respectively.
