This research delves into the problem of interactive editing of human motion generation. Previous motion diffusion models lack explicit modeling of the word-level text-motion correspondence and good explainability, hence restricting their fine-grained editing ability. To address this issue, we propose an attention-based motion diffusion model, namely MotionCLR, with CLeaR modeling of attention mechanisms. Technically, MotionCLR models the in-modality and cross-modality interactions with self-attention and cross-attention, respectively. More specifically, the self-attention mechanism aims to measure the sequential similarity between frames and impacts the order of motion features. By contrast, the cross-attention mechanism works to find the fine-grained word-sequence correspondence and activate the corresponding timesteps in the motion sequence. Based on these key properties, we develop a versatile set of simple yet effective motion editing methods via manipulating attention maps, such as motion (de-)emphasizing, in-place motion replacement, and example-based motion generation, etc. For further verification of the explainability of the attention mechanism, we additionally explore the potential of action-counting and grounded motion generation ability via attention maps. Our experimental results show that our method enjoys good generation and editing ability with good explainability.