Stylized motion generation is actively studied in computer graphics, especially benefiting from the rapid advances in diffusion models. The goal of this task is to produce a novel motion respecting both the motion content and the desired motion style, e.g., ``walking in a loop like a Monkey''. Existing research attempts to address this problem via motion style transfer or conditional motion generation. They typically embed the motion style into a latent space and guide the motion implicitly in a latent space as well. Despite the progress, their methods suffer from low interpretability and control, limited generalization to new styles, and fail to produce motions other than ``walking'' due to the strong bias in the public stylization dataset. In this paper, we propose to solve the stylized motion generation problem from a new perspective of reasoning-composition-generation, based on our observations: i) human motion can often be effectively described using natural language in a body-part centric manner, ii) LLMs exhibit a strong ability to understand and reason about human motion, and iii) human motion has an inherently compositional nature, facilitating the new motion content or style generation via effective recomposing. We thus propose utilizing body-part text space as an intermediate representation, and present SMooGPT, a fine-tuned LLM, acting as a reasoner, composer, and generator when generating the desired stylized motion. Our method executes in the body-part text space with much higher interpretability, enabling fine-grained motion control, effectively resolving potential conflicts between motion content and style, and generalizes well to new styles thanks to the open-vocabulary ability of LLMs. Comprehensive experiments and evaluations, and a user perceptual study, demonstrate the effectiveness of our approach, especially under the pure text-driven stylized motion generation.