Unmanned Aerial Vehicle (UAV) Vision-and-Language Navigation (VLN) is vital for applications such as disaster response, logistics delivery, and urban inspection. However, existing methods often struggle with insufficient multimodal fusion, weak generalization, and poor interpretability. To address these challenges, we propose FlightGPT, a novel UAV VLN framework built upon Vision-Language Models (VLMs) with powerful multimodal perception capabilities. We design a two-stage training pipeline: first, Supervised Fine-Tuning (SFT) using high-quality demonstrations to improve initialization and structured reasoning; then, Group Relative Policy Optimization (GRPO) algorithm, guided by a composite reward that considers goal accuracy, reasoning quality, and format compliance, to enhance generalization and adaptability. Furthermore, FlightGPT introduces a Chain-of-Thought (CoT)-based reasoning mechanism to improve decision interpretability. Extensive experiments on the city-scale dataset CityNav demonstrate that FlightGPT achieves state-of-the-art performance across all scenarios, with a 9.22% higher success rate than the strongest baseline in unseen environments. Our implementation is publicly available.