Aerial reconfigurable intelligent surfaces (ARIS), deployed on unmanned aerial vehicles (UAVs), could enhance anti-jamming communication performance by dynamically configuring channel conditions and establishing reliable air-ground links. However, large-scale ARIS faces critical deployment challenges due to the prohibitive computational complexity of conventional discrete optimization methods and sophisticated jamming threats. In this paper, we introduce a mean field modeling approach to design the spatial configuration of ARIS by a continuous density function, thus bypassing high-dimensional combinatorial optimization. We consider an adaptive jammer which adjusts its position and beamforming to minimize the sum-rate. A key finding reveals that the jammer's optimal strategy is governed by a proximity-directivity trade-off between reducing path loss and enhancing spatial focusing. To combat the jamming, we propose a robust anti-jamming transmission framework that jointly optimizes the BS beamforming, the ARIS reflection, and the ARIS spatial distribution to maximize the worst-case sum-rate. By leveraging variational optimization and Riemannian manifold methods, we efficiently solve the functional optimization problems. Our analysis further unveils that the optimal ARIS deployment follows a spatial water-filling principle, concentrating resources in high-gain regions while avoiding interference-prone areas. Simulation results demonstrate that the proposed framework remarkably improves the sum-rate. Furthermore, the computational complexity of the proposed algorithm is independent of the number of UAVs, validating its effectiveness for scalable ARIS-assisted anti-jamming communications.