Dimensionality reduction techniques are widely used for visualizing high-dimensional data. However, support for interpreting patterns of dimension reduction results in the context of the original data space is often insufficient. Consequently, users may struggle to extract insights from the projections. In this paper, we introduce DimBridge, a visual analytics tool that allows users to interact with visual patterns in a projection and retrieve corresponding data patterns. DimBridge supports several interactions, allowing users to perform various analyses, from contrasting multiple clusters to explaining complex latent structures. Leveraging first-order predicate logic, DimBridge identifies subspaces in the original dimensions relevant to a queried pattern and provides an interface for users to visualize and interact with them. We demonstrate how DimBridge can help users overcome the challenges associated with interpreting visual patterns in projections.