Multimodal data has significantly advanced recommendation systems by integrating diverse information sources to model user preferences and item characteristics. However, these systems often struggle with redundant and irrelevant information, which can degrade performance. Most existing methods either fuse multimodal information directly or use rigid architectural separation for disentanglement, failing to adequately filter noise and model the complex interplay between modalities. To address these challenges, we propose a novel framework, the Multimodal Representation-disentangled Information Bottleneck (MRdIB). Concretely, we first employ a Multimodal Information Bottleneck to compress the input representations, effectively filtering out task-irrelevant noise while preserving rich semantic information. Then, we decompose the information based on its relationship with the recommendation target into unique, redundant, and synergistic components. We achieve this decomposition with a series of constraints: a unique information learning objective to preserve modality-unique signals, a redundant information learning objective to minimize overlap, and a synergistic information learning objective to capture emergent information. By optimizing these objectives, MRdIB guides a model to learn more powerful and disentangled representations. Extensive experiments on several competitive models and three benchmark datasets demonstrate the effectiveness and versatility of our MRdIB in enhancing multimodal recommendation.