Low-altitude economy (LAE) represents an emerging economic paradigm that redefines commercial and social aerial activities. Large artificial intelligence models (LAIMs) offer transformative potential to further enhance the intelligence of LAE services. However, deploying LAIMs in LAE poses several challenges, including the significant gap between their computational/storage demands and the limited onboard resources of LAE entities, the mismatch between lab-trained LAIMs and dynamic physical environments, and the inefficiencies of traditional decoupled designs for sensing, communication, and computation. To address these issues, we first propose a hierarchical system architecture tailored for LAIM deployment and present representative LAE application scenarios. Next, we explore key enabling techniques that facilitate the mutual co-evolution of LAIMs and low-altitude systems, and introduce a task-oriented execution pipeline for scalable and adaptive service delivery. Then, the proposed framework is validated through real-world case studies. Finally, we outline open challenges to inspire future research.