Handling spatial references in natural language is a key challenge in tasks like autonomous navigation and robotic manipulation. Recent work has investigated various neural architectures for learning multi-modal representations of spatial concepts that generalize well across a variety of observations and text instructions. In this work, we develop accurate models for understanding spatial references in text that are also robust and interpretable. We design a text-conditioned relation network whose parameters are dynamically computed with a cross-modal attention module to capture fine-grained spatial relations between entities. Our experiments across three different prediction tasks demonstrate the effectiveness of our model compared to existing state-of-the-art systems. Our model is robust to both observational and instructional noise, and lends itself to easy interpretation through visualization of intermediate outputs.