Rapid, accurate and robust detection of looming objects in cluttered moving backgrounds is a significant and challenging problem for robotic visual systems to perform collision detection and avoidance tasks. Inspired by the neural circuit of elementary motion vision in the mammalian retina, this paper proposes a bioinspired approach-sensitive neural network (ASNN) that contains three main contributions. Firstly, a direction-selective visual processing module is built based on the spatiotemporal energy framework, which can estimate motion direction accurately via only two mutually perpendicular spatiotemporal filtering channels. Secondly, a novel approach-sensitive neural network is modeled as a push-pull structure formed by ON and OFF pathways, which responds strongly to approaching motion while insensitivity to lateral motion. Finally, a method of directionally selective inhibition is introduced, which is able to suppress the translational backgrounds effectively. Extensive synthetic and real robotic experiments show that the proposed model is able to not only detect collision accurately and robustly in cluttered and dynamic backgrounds but also extract more collision information like position and direction, for guiding rapid decision making.