We propose the Gradient-weighted Object Detector Activation Mapping (Grad-ODAM), a visualized explanation technique for interpreting the predictions of object detectors. Utilizing the gradients of detector targets flowing into the intermediate feature maps, Grad-ODAM produces heat maps that show the influence of regions on the detector's decision. Compared to previous classification activation mapping works, Grad-ODAM generates instance-specific explanations rather than class-specific ones. We show that Grad-ODAM is applicable to both one-stage detectors such as FCOS and two-stage detectors such as Faster R-CNN, and produces higher-quality visual explanations than the state-of-the-art both effectively and efficiently. We next propose a training scheme, ODAM-Train, to improve the explanation ability on object discrimination of the detector through encouraging consistency between explanations for detections on the same object, and distinct explanations for detections on different objects. Based on the heat maps produced by Grad-ODAM with ODAM-Train, we propose ODAM-NMS, which considers the information of the model's explanation for each prediction to distinguish the duplicate detected objects. We present a detailed analysis of the visualized explanations of detectors and carry out extensive experiments to validate the effectiveness of the proposed ODAM.