Edible films based on biopolymers are used to protect food from adverse environmental factors. However, their ample use may be hindered by some challenges to their mechanical and antimicrobial properties. Despite this, in most cases, increasing their mechanical properties and antibacterial activity remains a relevant challenge. To solve this problem, a possible option is to fill the biopolymer matrix of films with a functional filler that combines high reinforcing and antibacterial properties. In this work, biocomposite films based on a mixture of chitosan and cassava starch were filled with a hybrid filler in the form of bentonite clay particles loaded with ginger essential oil (GEO) in their structure with varied concentrations. For this purpose, GEO components were intercalated into bentonite clay interlayer space using a mechanical capture approach without using surface-active and toxic agents. The structure and loading efficiency of the essential oil in the obtained hybrid filler were analyzed by lyophilization and laser analysis of dispersions, ATR-FTIR spectroscopy, thermogravimetry, and X-ray diffraction analysis. The filled biocomposite films were analyzed using ATR-FTIR spectroscopy, optical and scanning electron spectroscopy, energy dispersive spectroscopy, mechanical analysis under tension, and the disk diffusion method for antibacterial activity. The results demonstrated that the tensile strength, Young’s modulus, elongation at the break, and the antibacterial effect of the films increased by 40%, 19%, 44%, and 23%, respectively, compared to unfilled film when the filler concentration was 0.5–1 wt.%.