The integration of agentic AI, powered by large language models (LLMs) with autonomous reasoning, planning, and execution, into unmanned aerial vehicle (UAV) swarms opens new operational possibilities and brings the vision of the Internet of Drones closer to reality. However, infrastructure constraints, dynamic environments, and the computational demands of multi-agent coordination limit real-world deployment in high-risk scenarios such as wildfires and disaster response. This paper investigates the integration of LLM-based agentic AI and edge computing to realize scalable and resilient autonomy in UAV swarms. We first discuss three architectures for supporting UAV swarms - standalone, edge-enabled, and edge-cloud hybrid deployment - each optimized for varying autonomy and connectivity levels. Then, a use case for wildfire search and rescue (SAR) is designed to demonstrate the efficiency of the edge-enabled architecture, enabling high SAR coverage, reduced mission completion times, and a higher level of autonomy compared to traditional approaches. Finally, we highlight open challenges in integrating LLMs and edge computing for mission-critical UAV-swarm applications.