To "remain vigilant" means to stay alert, cautious and watchful for potential dangers or threats. It involves being constantly aware of your surroundings and taking necessary precautions to ensure safety. The phrase is often used in the context of security measures, where one must remain vigilant to prevent any untoward incidents. In a broader sense, it also means being attentive and cautious in all aspects of life, so as to avoid any potential harm or loss.