Despite the impressive performance on information-seeking tasks, large language models (LLMs) still struggle with hallucinations. Attributed LLMs, which augment generated text with in-line citations, have shown potential in mitigating hallucinations and improving verifiability. However, current approaches suffer from suboptimal citation quality due to their reliance on in-context learning. Furthermore, the practice of citing only coarse document identifiers makes it challenging for users to perform fine-grained verification. In this work, we introduce FRONT, a training framework designed to teach LLMs to generate Fine-Grained Grounded Citations. By grounding model outputs in fine-grained supporting quotes, these quotes guide the generation of grounded and consistent responses, not only improving citation quality but also facilitating fine-grained verification. Experiments on the ALCE benchmark demonstrate the efficacy of FRONT in generating superior grounded responses and highly supportive citations. With LLaMA-2-7B, the framework significantly outperforms all the baselines, achieving an average of 14.21% improvement in citation quality across all datasets, even surpassing ChatGPT.