Low Earth orbit (LEO) mega-constellations, such as Starlink and Kuiper, are revolutionizing global connectivity by providing internet access to remote and oceanic regions. To meet explosive transmission demands for 6G applications, including the metaverse, the Internet of Things (IoT), and holographic communication, hybrid freespace optical (FSO) and radio frequency (RF) links offer a promising solution. FSO provides ultra-high throughput, and RF offers weather-resilient reliability. To address these conflicts, we introduce an intelligent, user-centric connectivity architecture for hybrid FSO/RF LEO satellite networks, leveraging software-defined satellite networking (SDSN) and collaborative deep reinforcement learning (DRL). We guide readers through entire connection lifecycles, key technical challenges, and an intelligent solution that enables high-quality service in dynamic space environments. A case study illustrates intelligent orchestration in a mega-constellation environment, highlighting the trade-offs between switching frequency and link stability. Concurrently, we explore development directions for intelligent 6G Space-Air-Ground Integrated Networks (SAGIN).