An influential framework within systems neuroscience posits that neural computations can be understood in terms of low-dimensional dynamics in recurrent circuits. A number of methods have thus been developed to extract latent dynamical systems from neural recordings, but inferring models that are both predictive and interpretable remains a difficult challenge. Here we propose a new method called Low-rank Inference from Neural Trajectories (LINT), based on a class of low-rank recurrent neural networks (lrRNNs) for which a link between connectivity and dynamics has been previously demonstrated. By fitting such networks to trajectories of neural activity, LINT yields a mechanistic model of latent dynamics, as well as a set of axes for dimensionality reduction and verifiable predictions for inactivations of specific populations of neurons. Here, we first demonstrate the consistency of our method and apply it to two use cases: (i) we reverse-engineer "black-box" vanilla RNNs trained to perform cognitive tasks, and (ii) we infer latent dynamics and neural contributions from electrophysiological recordings of nonhuman primates performing a similar task.