(Note: $c =1$ throughout) The Schwarzschild metric is $$ds^2 = (1- \frac{2m}{r})dt^2 - \frac{1}{1-\frac{2m}{r}}dr^2 - r^2 d\Omega ^2,$$ with $d\Omega^2$ being the square of the solid angle element and $m = GM$, where $M$ is the mass of the object. Radial null geodesics in this geometry are given by $$t_\pm(r) = \pm(r+2m\log |r-2m|+C),$$ where $r = \pm k\lambda$, $\lambda$ an affine parameter, with the plus sign indicating that the particle is outgoing, and the minus sign indicating that it is ingoing. My question is: what does this physically represent? After thinking about it for some time, I have considered that it might be the time taken for a light particle to reach a distance $r$ from the singularity, but where is it falling from? If anyone could clarify my confusions, it would be greatly appreciated.
Edit: The photon would be falling from an initial position $r_0$, where $\mp (r_0 + \log|r_0 -2m|) = C$ as mentioned by Triatticus and myself.