Collaborative multiple robots for unknown environment exploration have become mainstream due to their remarkable performance and efficiency. However, most existing methods assume perfect robots' communication during exploration, which is unattainable in real-world settings. Though there have been recent works aiming to tackle communication-constrained situations, substantial room for advancement remains for both information-sharing and exploration strategy aspects. In this paper, we propose a Communication-Constrained Multi-Robot Entropy-Field-Based Exploration (MEF-Explore). The first module of the proposed method is the two-layer inter-robot communication-aware information-sharing strategy. A dynamic graph is used to represent a multi-robot network and to determine communication based on whether it is low-speed or high-speed. Specifically, low-speed communication, which is always accessible between every robot, can only be used to share their current positions. If robots are within a certain range, high-speed communication will be available for inter-robot map merging. The second module is the entropy-field-based exploration strategy. Particularly, robots explore the unknown area distributedly according to the novel forms constructed to evaluate the entropies of frontiers and robots. These entropies can also trigger implicit robot rendezvous to enhance inter-robot map merging if feasible. In addition, we include the duration-adaptive goal-assigning module to manage robots' goal assignment. The simulation results demonstrate that our MEF-Explore surpasses the existing ones regarding exploration time and success rate in all scenarios. For real-world experiments, our method leads to a 21.32% faster exploration time and a 16.67% higher success rate compared to the baseline.