Transporting a heavy payload using multiple aerial robots (MARs) is an efficient manner to extend the load capacity of a single aerial robot. However, existing schemes for the multiple aerial robots transportation system (MARTS) still lack the capability to generate a collision-free and dynamically feasible trajectory in real-time and further track an agile trajectory especially when there are no sensors available to measure the states of payload and cable. Therefore, they are limited to low-agility transportation in simple environments. To bridge the gap, we propose complete planning and control schemes for the MARTS, achieving safe and agile aerial transportation (SAAT) of a cable-suspended payload in complex environments. Flatness maps for the aerial robot considering the complete kinematical constraint and the dynamical coupling between each aerial robot and payload are derived. To improve the responsiveness for the generation of the safe, dynamically feasible, and agile trajectory in complex environments, a real-time spatio-temporal trajectory planning scheme is proposed for the MARTS. Besides, we break away from the reliance on the state measurement for both the payload and cable, as well as the closed-loop control for the payload, and propose a fully distributed control scheme to track the agile trajectory that is robust against imprecise payload mass and non-point mass payload. The proposed schemes are extensively validated through benchmark comparisons, ablation studies, and simulations. Finally, extensive real-world experiments are conducted on a MARTS integrated by three aerial robots with onboard computers and sensors. The result validates the efficiency and robustness of our proposed schemes for SAAT in complex environments.