 Solution

The emission term comes from the local photon temperature. A blackbody radiation density is proportional to $T^4$, so linearization gives $\delta T/T={\delta }_{\gamma }/4$. It is evaluated at last scattering, with the source position on the observed ray.

The velocity term is the Doppler CMB anisotropy: motion of the emitting plasma changes the measured photon frequency by the component of its peculiar velocity along the line of sight. Its sign here follows the paper's choice of the sky direction $\hat{\mathbf{n}}$.

The metric integral follows the null geodesic energy equation in synchronous gauge. Time dependence of the spatial metric perturbation changes the photon energy by $-h_{ij}^{\prime }{n}_i{n}_j/2$ along the ray. It includes gravitational redshift endpoint terms as well as the part due to evolving gravitational potentials. In particular, the entire synchronous metric integral is not just the integrated Sachs-Wolfe effect. Integrating by parts to express it in conformal Newtonian gauge separates the ordinary Sachs-Wolfe effect from the evolving-potential contribution.