We propose a high-capacity, high-speed demodulation system for fiber Bragg grating (FBG) sensor arrays using dispersion Fourier transform technology. The proposed system includes a mode-locked laser, a nonlinear amplifier module, a sensing link, a dispersive element, and a signal acquisition and processing module. The sensor capacity of the system is substantially improved by employing a nonlinear amplifier to broaden the spectrum of the mode-locked laser, enabling high-speed multiplexing and demodulation of FBG sensing arrays. The temperature demodulation experiments demonstrate that the proposed system offers high demodulation accuracy and stability, simultaneously demodulating 38 FBG sensors in real time at a demodulation rate of 16.24 MHz.