Cross-speaker style transfer is crucial to the applications of multi-style and expressive speech synthesis at scale. It does not require the target speakers to be experts in expressing all styles and to collect corresponding recordings for model training. However, the performances of existing style transfer methods are still far behind real application needs. The root causes are mainly twofold. Firstly, the style embedding extracted from single reference speech can hardly provide fine-grained and appropriate prosody information for arbitrary text to synthesize. Secondly, in these models the content/text, prosody, and speaker timbre are usually highly entangled, it's therefore not realistic to expect a satisfied result when freely combining these components, such as to transfer speaking style between speakers. In this paper, we propose a cross-speaker style transfer text-to-speech (TTS) model with explicit prosody bottleneck. The prosody bottleneck builds up the kernels accounting for speaking style robustly, and disentangles the prosody from content and speaker timbre, therefore guarantees high quality cross-speaker style transfer. Evaluation result shows the proposed method even achieves on-par performance with source speaker's speaker-dependent (SD) model in objective measurement of prosody, and significantly outperforms the cycle consistency and GMVAE-based baselines in objective and subjective evaluations.