Large language models (LLMs) are now at the core of conversational AI services such as real-time translation and chatbots, which provide live user interaction by incrementally streaming text to the user. However, existing LLM serving systems fail to provide good user experience because their optimization metrics are not always aligned with user experience. In this paper, we first introduce and define the notion of Quality-of-Experience (QoE) for text streaming services by considering each user's end-to-end interaction timeline. Based on this, we propose Andes, a QoE-aware LLM serving system that enhances user experience by ensuring that users receive the first token promptly and subsequent tokens at a smooth, digestible pace, even during surge periods. This is enabled by Andes's preemptive request scheduler that dynamically prioritizes requests at the token granularity based on each request's expected QoE gain and GPU resource usage. Our evaluations demonstrate that, compared to state-of-the-art LLM serving systems, Andes improves the average QoE by up to $4.7\times$ given the same GPU resource, or saves up to 61% GPU resources while maintaining the same high QoE.