Embodied agents, which couple intelligent decision-making with physical actuation in the real world, impose far more stringent and heterogeneous communication requirements than purely software-based agents. While 6G promises sub-millisecond latency, ultra-high reliability, native intelligence, and integrated sensing, systematic studies on how to exploit these capabilities for embodied agent communication remain limited. This article investigates 6G-enabled communication systems for embodied agents from both conceptual and engineering perspectives. First, we review the concept, embodiment value of embodied agents, and clarify their distinctions from disembodied agents. Then, we analyse the symbiotic relationship between embodied agents and 6G networks. We highlight how key 6G enablers can support the stringent requirements of human-robot interaction. Furthermore, we demonstrate the proactive role of embodied agents in bolstering communication networks through coverage extension, environmental sensing, and physical world understanding. Building on these insights, we propose a hierarchical communication architecture for human-robot remote interaction, comprising a human-intent perception layer, an open radio access network (O-RAN)-based transport layer, an intelligent intermediary layer, and an embodiment layer. To validate its feasibility, we implement an end-to-end prototype that integrates a haptic device, an industrial robotic arm, an intermediary platform, and a 5G O-RAN testbed. Experimental results demonstrate millisecond-level latency and stable closed-loop operation, confirming the practicality of the proposed architecture and providing a reference for future 6G-embodied agent research and industrial deployments.