We present the temporal evolution of the longitudinal spin Seebeck effect in a YIG|Pt bilayer system. Our findings reveal that this effect is a submicrosecond fast phenomenon governed by the thermal-magnon diffusion along the thermal gradient inside the magnetic material. A comparison of experimental results with the thermal-driven magnon-diffusion model demonstrates that the temporal behavior of this effect depends on the time development of the temperature gradient in the magnetic material close to the interface. The effective thermal-magnon diffusion length for the YIG|Pt system is estimated to be around 500 nm.