Resource management in sixth-generation (6G) space-air–ground integrated network (SAGINs) is becoming increasingly challenging due to cross-tier heterogeneity, multi-timescale dynamics, and distributed decision making under partial and delayed information. These characteristics expose the limitations of conventional optimization-based and learning-based approaches, whose effectiveness often depends on accurate modeling, large amounts of task-specific data, or opaque policy adaptation. This article revisits the evolution of SAGIN resource management from optimization and learning to reasoning-centric, large language model (LLM)-enabled agentic intelligence. A unified taxonomy of existing paradigms is first presented, clarifying their respective strengths and limitations. The operational foundations of LLM-enabled agents (LEAs) are then introduced, including perception, memory, thought, execution, and tool-grounded decision making, followed by a discussion of emerging multi-agent frameworks for distributed orchestration. Building on this foundation, LEAs are shown to support proactive spectrum control, autonomous cross-tier routing, multi-service scheduling, and dynamic network slicing. Finally, key open challenges are identified, and future directions toward practical, trustworthy, and scalable agentic resource management in 6G SAGINs are outlined.