Transversal magneto-optical Kerr effect (TMOKE) has potential practical applications, such as biosensors, magnetic imaging, and date storage. However, these potential applications have been restricted by its very weak response (about 0.1%) in natural ferromagnetic metal material such as Fe, Co and Ni. Fortunately, with the development of the nanofabrication techniques, surface plasmons (SPs) are one of the effective strategies to solve this problem due to their special ability to manipulate light on a nanoscale and concentrate the electromagnetic energy near the metal/dielectric interface. Herein, in order to enhance the TMOKE response, we propose that a periodic gold strips array is embedded into a magnetic dielectric film of bismuth iron garnet (BIG), which is supported by a quartz substrate. Using the finite element method, we numerically study the optical properties of our proposed microstructure and the corresponding evolution of the TMOKE responses due to the coupled optical modes dependent on the structural parameters. Particularly, by optimizing the embedded depth of metal grating, a dramatic enhancement of TMOKE response (about 3.6%) is achieved when the embedded depth reaches up to 80 nm, accompanied with a high transmissivity about 22.6%, which is actually three time larger than that in the case that the gold strips are just patterned on the surface of the BIG film. As the embedding depth increases further, the TMOKE response will be weak. The relationship between the TMOKE response and the coupling efficiency of LSP resonance of the gold stripes and the waveguide (WG) mode supported by the BIG film are also discussed systematically. As the embedding depth increases up to 80 nm gradually, the coupling of the WG mode in BIG film with the LSP mode of the individual gold stripe becomes much stronger and forms a highly efficient Fano resonance, which leads to the fact that most of the electromagnetic field is localized in the BIG film and strong interaction with the BIG magnetic dielectric film, and thus, an enhancement of TMOKE response can be observed. However, when the embedded depth increases further, the uniformity of BIG film will be broken. In this case, WG mode cannot be supported by BIG film very well any more at the wavelength corresponding to excitation of the LSP, which results in a weakly coupling efficiency between LSP and WG mode. In this case, the Fano resonance cannot be formed and rare electromagnetic field can be localized in the BIG film, leading to a very weak light-magnetic dielectric film interaction and the weak TMOKE response. Our study proposes a new method to realize the amplification of weak TMOKE response by utilizing the plasmonic microstructure, which might have a potential application to designing the high-efficiency magneto-optical devices.