The inherent noisy and sparse characteristics of radar data pose challenges in finding effective representations for 3D object detection. In this paper, we propose RadarDistill, a novel knowledge distillation (KD) method, which can improve the representation of radar data by leveraging LiDAR data. RadarDistill successfully transfers desirable characteristics of LiDAR features into radar features using three key components: Cross-Modality Alignment (CMA), Activation-based Feature Distillation (AFD), and Proposal-based Feature Distillation (PFD). CMA enhances the density of radar features through multiple layers of dilation operations, effectively addressing the challenges of inefficient knowledge transfer from LiDAR to radar. AFD is designed to transfer knowledge from significant areas of the LiDAR features, specifically those regions where activation intensity exceeds a predetermined threshold. PFD guides the radar network to mimic LiDAR network features in the object proposals for accurately detected results while moderating features for misdetected proposals like false positives. Our comparative analyses conducted on the nuScenes datasets demonstrate that RadarDistill achieves state-of-the-art (SOTA) performance for radar-only object detection task, recording 20.5% in mAP and 43.7% in NDS. Also, RadarDistill significantly improves the performance of the camera-radar fusion model.