For cyber-physical systems in the 6G era, semantic communications connecting distributed devices for dynamic control and remote state estimation are required to guarantee application-level performance, not merely focus on communication-centric performance. Semantics here is a measure of the usefulness of information transmissions. Semantic-aware transmission scheduling of a large system often involves a large decision-making space, and the optimal policy cannot be obtained by existing algorithms effectively. In this paper, we first investigate the fundamental properties of the optimal semantic-aware scheduling policy and then develop advanced deep reinforcement learning (DRL) algorithms by leveraging the theoretical guidelines. Our numerical results show that the proposed algorithms can substantially reduce training time and enhance training performance compared to benchmark algorithms.