Cyber-physical systems (CPSs) are increasingly threatened by stealthy false data injection (SFDI) attacks, which compromise system integrity by manipulating control signals and introducing false sensor data. These attacks are particularly challenging due to their diversity and often indistinguishable nature. In response to this issue, our work uncovers the fundamental causes behind SFDI attacks in linear time-invariant (LTI) systems and elucidates the principles enabling their stealth. We present a novel virtual extended system framework designed to eliminate strictly stealthy attacks within the entire CPS. Utilizing deep reinforcement learning (DRL) methodologies, we pioneer the use of detection results for real-time SFDI attack classification. Through numerical simulations, we validate our proposed method's effectiveness, demonstrating a classification accuracy of no less than 95%. Notably, even in scenarios where attackers manage to breach the framework partially, our method continues to provide a reliable success rate in SFDI attack detection and classification, showcasing its robustness and efficacy. Note to Practitioners -Cyber-physical systems (CPSs), a critical component of modern industries, are becoming increasingly susceptible to stealthy false data injection (SFDI) attacks. These attacks compromise system integrity by subtly manipulating control signals and feeding false sensor data, making them challenging to detect. Our research presents an innovative framework that uses deep reinforcement learning techniques to detect and classify these elusive attacks, achieving a classification accuracy of over 95%. The information on SFDI attack categories, ascertained by this method, lays the groundwork for the development of subsequent defence strategies. For professionals working in sectors reliant on CPSs, such as manufacturing, healthcare, and transportation, this framework offers a promising tool to enhance system security. Even in scenarios where the system has been partially compromised, our method continues to provide reliable detection and classification, underscoring its robustness and practical utility. The system remains effective despite full breach attempts on specific attack types, ensuring resilience against a broad range of SFDI attacks. In conclusion, our research offers a substantial advancement in protecting CPSs against cyber threats.