Page 248 gives us this action and he simply says that we will assume it correct.
$$ S=\int d \tau d \sigma ~\mathcal{L}=\frac{1}{4 \pi \alpha^{\prime}} \int d \tau \int_{0}^{\pi} d \sigma\left(\dot{X}^{I} \dot{X}^{I}-X^{I^{\prime}} X^{I^{\prime}}\right),\tag{12.81} $$ where $X^I$ denote transverse target-space coordinates.
Besides giving us the right answer at the end, what is the motivation for this action, how was it thought up? It seems like a modified Nambu-Goto action.
FWIW, it is in principle possible to systematically derive the light-cone action (12.81) from (the Hamiltonian formulation of) the Nambu-Goto (NG) string, see e.g. this related Phys.SE post.
The NG action is in principle equivalent to the Polyakov action, cf. e.g. this Phys.SE post.
In both cases, it takes a bit of work to consistently reduce to the transverse degrees of freedom of the light-cone formulation (12.81).
It's the simplest action of a field $X^I(\sigma,\tau)$ defined on a 2d worldsheet which is manifestly Lorentz invariant in target space. Your action is a special case of the Polyakov action where both the worldsheet metric $h^{ab}(\sigma,\tau)$ and the target space metric $g_{\mu \nu}(X)$ are taken to be flat.
$$ \delta S=\int_{\tau_{i}}^{\tau_{f}} d \tau\left[\delta X^{\mu} \mathcal{P}_{\mu}^{\sigma}\right]_{0}^{\sigma_{1}}-\int_{\tau_{i}}^{\tau_{f}} d \tau \int_{0}^{\sigma_{1}} d \sigma \delta X^{\mu}\left(\frac{\partial \mathcal{P}_{\mu}^{\tau}}{\partial \tau}+\frac{\partial \mathcal{P}_{\mu}^{\sigma}}{\partial \sigma}\right) $$
$$ \frac{\partial \mathcal{L}}{\partial \dot{X}^{I}}=\frac{1}{2 \pi \alpha^{\prime}} \dot{X}^{I}=\mathcal{P}^{\tau I} $$
Then working in reverse to find the action. I'm not sure if it's cyclical or not but I don't like going the Hamiltonian way for the exact same reason.
