Reconstructing continuous flow fields from sparse surface-mounted sensors is central to aerodynamic design, flow control, and digital-twin instrumentation. Existing neural methods for this task typically encode sensor readings into implicit latent codes with little spatial interpretability and limited formal guidance on how representational capacity should scale with observation count. Inspired by 3D Gaussian Splatting, we introduce FLUIDSPLAT, a sensor-conditioned model that predicts K anisotropic Gaussian primitives forming a partition-of-unity scaffold, a spatially explicit and interpretable intermediate representation of the flow. For an idealized Gaussian primitive estimator, we prove an $O(K^{-s/d})$ approximation rate for fields with Sobolev smoothness $s$; incorporating $N$ noisy observations yields a squared-risk decomposition with bias $O(K^{-2s/d})$ and variance $O(σ^{2}K/N)$.Balancing the two yields $K^{*}\!\sim\!(N/σ^{2})^{d/(2s+d)}$: primitive count cannot grow freely under sparse sensing, revealing a variance bottleneck that motivates complementing the scaffold with a state-conditioned residual decoder. Across four benchmarks spanning 2D and 3D, FLUIDSPLAT achieves 11-28% error reduction over several strong baselines on cylinder flow, AirfRANS, FlowBench LDC-3D, and PhySense-Car 3D benchmarks.