Because it is the natural time scale of the problem. When you have a damped harmonic oscillator then its amplitude decreases with time like ##e^{-t/\tau}##. In this case ##\tau## is the time scale that determines the behavior of the system: if ##t\ll\tau## the system is still oscillating as an ordinary harmonic oscillator, while for ##t\gg\tau## it is already completely damped. This is why ##t=\tau## (the relaxation time) determines some sort of "benchmark" in the state of the system.