Neural network-based driving planners have shown great promises in improving task performance of autonomous driving. However, it is critical and yet very challenging to ensure the safety of systems with neural network based components, especially in dense and highly interactive traffic environments. In this work, we propose a safety-driven interactive planning framework for neural network-based lane changing. To prevent over conservative planning, we identify the driving behavior of surrounding vehicles and assess their aggressiveness, and then adapt the planned trajectory for the ego vehicle accordingly in an interactive manner. The ego vehicle can proceed to change lanes if a safe evasion trajectory exists even in the predicted worst case; otherwise, it can stay around the current lateral position or return back to the original lane. We quantitatively demonstrate the effectiveness of our planner design and its advantage over baseline methods through extensive simulations with diverse and comprehensive experimental settings, as well as in real-world scenarios collected by an autonomous vehicle company.