Context. Solar active region (AR) 12673 in 2017 September produced the two largest flares in Solar Cycle 24: the X9.3 flare on September 6 and the X8.2 flare on September 10. Aims. We attempt to investigate the evolutions of the two large flares and their associated complex magnetic system in detail. Methods. Combining observations from the Solar Dynamics Observatory and results of nonlinear force-free field (NLFFF) modeling, we identify various magnetic structures in the AR core region and examine the evolution of these structures during the flares. Results. Aided by the NLFFF modeling, we identify a double-decker flux rope configuration above the polarity inversion line (PIL) in the AR core region. The north ends of these two flux ropes were rooted in a negative- polarity magnetic patch, which began to move along the PIL and rotate anticlockwise before the X9.3 flare on September 6. The strong shearing motion and rotation contributed to the destabilization of the two magnetic flux ropes, of which the upper one subsequently erupted upward due to the kink-instability. Then another two sets of twisted loop bundles beside these ropes were disturbed and successively erupted within five minutes like a chain reaction. Similarly, multiple ejecta components were detected as consecutively erupting during the X8.2 flare occurring in the same AR on September 10. We examine the evolution of the AR magnetic fields from September 3 to 6 and find that five dipoles emerged successively at the east of the main sunspot. The interactions between these dipoles took place continuously, accompanied by magnetic flux cancellations and strong shearing motions. Conclusions. In AR 12673, significant flux emergence and successive interactions between the different emerging dipoles resulted in a complex magnetic system, accompanied by the formations of multiple flux ropes and twisted loop bundles. We propose that the eruptions of a multi-flux-rope system resulted in the two largest flares in Solar Cycle 24.