Dispersive Fourier transform offers a powerful real-time way to measure the transient, complex, and non-repetitive nonlinear dynamics of the soliton in a passively mode-locked fiber laser. With this technique, we investigate the real-time fission dynamics of a dissipative soliton in a passively mode-locked fiber laser based on the nonlinear polarization rotation technique. The results show that the dissipative soliton fissions occur when the pump power is 280 mW and that a single soliton pulse is generated by the background noise pulse before the dissipative soliton fissions. Different from the single soliton mode-locking state when the pump power is 180 mW, a new spectral component is observed on the spectrum of the single soliton generated before fission. The new frequency component amplifies and stretches with the increase in the number of round trips and ultimately evolves into a new dissipative soliton pulse. This research is significant for understanding the formation and fission mechanism of the dissipative soliton in a passively mode-locked fiber laser.