Implicit communication is crucial in human-robot collaboration (HRC), where contextual information, such as intentions, is conveyed as implicatures, forming a natural part of human interaction. However, enabling robots to appropriately use implicit communication in cooperative tasks remains challenging. My research addresses this through three phases: first, exploring the impact of linguistic implicatures on collaborative tasks; second, examining how robots' implicit cues for backchanneling and proactive communication affect team performance and perception, and how they should adapt to human teammates; and finally, designing and evaluating a multi-LLM robotics system that learns from human implicit communication. This research aims to enhance the natural communication abilities of robots and facilitate their integration into daily collaborative activities.