The community integrated energy system (CIES) has emerged as a prominent solution for enhancing the flexibility of the distribution system with high renewable energy penetration by the seamless integration and coordination of heterogeneous energy sources and demand-side flexibility resources. However, with the escalating computational demands and massive data traffic of the energy internet, the limited computing resources at the centralized cloud engender substantial hurdles in holistic scheduling. Besides, the flexibility response potential of considerable resources in community users is constrained by multiple subjective factors. To this end, a cloudedge collaboration based real-time flexibility scheduling framework incorporating consumer psychology is proposed to accelerate the intellectualization and flexibility of CIES. Firstly, a foundational CIES model integrates electricity, heat, and natural gas networks is comprehensively established, implementing tiered utilization of diverse energy flows for synergies. Then, a cloud-edge collaboration hierarchical scheduling strategy is proposed to manage CIES. For the application layer, a demand-side hybrid load aggregation model is developed based on the load characteristics. Subsequentially, a coordinated control method incorporating the optimal task offloading strategy and hierarchical scheduling strategy is introduced for the distributed coordination layer and centralized control layer. Finally, the consumer psychology is investigated during the hierarchical scheduling process by modelling user behavior through the fuzzy response mechanism based on logistic function. The proposed approach optimizes the real-time scheduling of CIES by reducing system latency and improving demand-side flexibility, thereby lowering operational costs. Simulation results demonstrate a notable enhancement in flexibility provision, with upward and downward flexibility increasing by approximately 11.49 % and 11.93 %, respectively, compared to traditional real-time scheduling strategy. Furthermore, the integration of cloud-edge collaboration reduces transmission latency by 10.23 % and computation latency by 1.46 %, thereby improving scheduling efficiency. Besides, electricity price incentives and latency issues significantly influence user response willingness, necessitating comprehensive consideration in practical applications.