I) What does it mean that a theory is invariant wrt. a group $G$? Well, in particular this means that there is a well-defined given prescription on how the constituents of the theory change under the action of the group. Often in physics (but not always), it happens that the group action is linearly realized, i.e. the fields, the matrix elements, and other objects form linear representations of the group $G$. Let us for simplicity assume this in what follows. See also e.g. this and this Phys.SE post.
II) In our case the group $G$ is the Lorentz Lie group $O(D-1,1)$. This is the set of real $D\times D$ matrices $\Lambda\in {\rm Mat}_{D\times D}(\mathbb{R})$, such that
$$ \Lambda^t \eta \Lambda ~=~\eta.$$
Here $\eta_{\mu\nu}$ is the Minkowski metric in $D$ spacetime dimensions. Thus we want to study a linear representation $\rho$ of the Lorentz Lie group. Note that the representation $\rho(\Lambda)$ of a Lorentz matrix $\Lambda$ is typically not a $D\times D$ matrix. It could be an operator. Here the representation (map) $\rho$ is a Lie group homomorphism.
III) Then, at the infinitesimal level, the $D\times D$ Lorentz matrices
$$\Lambda~\in~ O(D-1,1)~\subseteq ~{\rm Mat}_{D\times D}(\mathbb{R}) $$
are generated by a $D(D-1)/2$-dimensional basis of $D\times D$ matrices $J^{\mu\nu}\equiv-J^{\nu\mu}$ in the corresponding Lie algebra
$$so(D-1,1)~\subseteq ~{\rm Mat}_{D\times D}(\mathbb{R}).$$
Similarly, at the infinitesimal level, the Lie group action on a physical object (in the theory) is generated by the corresponding Lie algebra generators $M^{\mu\nu}=\rho(J^{\mu\nu})$ in the pertinent linear representation $\rho$.
As a consequence of the group action, the Lorentz Lie algebra generators $M^{\mu\nu}$ in the relevant linear representation $\rho$ have to satisfy the explicit form of the $so(D-1,1)$ Lorentz Lie-algebra.
IV) Finally, use (i) this explicit form and (ii) the fact that the Minkowski metric in light-cone coordinates is given as
$$\eta_{\mu\nu}~=~\left[ \begin{array}{cc} 0 & -1 & 0 \\-1 & 0 & 0 \\ 0 & 0 &\delta_{IJ}\end{array} \right], \qquad \mu,\nu=+,-,2,3,\ldots, D-1,$$
$$\qquad I,J=2,3,\ldots, D-1,$$
to conclude that
$$[M^{I-},M^{J-}]~=~0.$$
V) At the classical level, one may use Noether's Theorem to find the classical representation of the angular momentum generators $M^{\mu\nu}$.
Concretely how the classical bosonic string is quantized, and concretely how the corresponding quantum operators $M^{\mu\nu}$ are explicitly realized as operators, is e.g. explained in Barton Zwiebach, A First Course in String Theory, Chapter 12. The construction is non-trivial due to operator ordering issues.
