Free-text rationales (FTRs) follow how humans communicate by explaining reasoning processes via natural language. A number of recent works have studied how to improve language model (LM) generalization by using FTRs to teach LMs the correct reasoning processes behind correct task outputs. These prior works aim to learn from FTRs by appending them to the LM input or target output, but this may introduce an input distribution shift or conflict with the task objective, respectively. We propose KNIFE, which distills FTR knowledge from an FTR-augmented teacher LM (takes both task input and FTR) to a student LM (takes only task input), which is used for inference. Crucially, the teacher LM's forward computation has a bottleneck stage in which all of its FTR states are masked out, which pushes knowledge from the FTR states into the task input/output states. Then, FTR knowledge is distilled to the student LM by training its task input/output states to align with the teacher LM's. On two question answering datasets, we show that KNIFE significantly outperforms existing FTR learning methods, in both fully-supervised and low-resource settings.