Low Earth Orbit Satellite Networks (LEOSNs) have emerged as a promising paradigm for space information networks, where multiple inter-satellite links facilitate the rapid transmission of on-orbit data to multi-ground station systems. When the destination of the transmission is not a certain ground station, but anyone of the ground stations, it can be modeled as an anycast problem. However, the time-varying topology, dynamic inter-satellite link status and limited onboard resources bring challenges to the routing of on-orbit data. Existing unicast routing solutions failed to address the anycast routing problem as they could not fully utilize multiple ground stations. Inspired by the Potential Field (PF) theory in physics, we make the first attempt to adopt the PF approach in the on-orbit data anycast routing problem. We design a synthesized PF model including several sub-fields corresponding to network states such as length of path, node transmission load and link bandwidth. By implementing inter-satellite propagation and synthesis of PF, dynamic perception and unified measurement of network states can be achieved. Based on our PF model, we propose a distributed PF-based and Network State-aware Anycast Routing (PFNSAR) algorithm, which regards the ground stations as multiple sources of attractive potential and guides packets along the gradient of synthesized PF. Meanwhile, the occurrence of the well-known local minimum is novelly handled by setting PF configuration subject to a parameter constraint and switching routing modes when routing packets. Extensive simulations demonstrate that PFNSAR provides unified assessment of multi-dimensional network states, reduces up to 30% of average delivery delay, and improves the performance including packet delivery rate and load balancing compared with existing works.