This paper explores the weakly-supervised referring image segmentation (WRIS) problem, and focuses on a challenging setup where target localization is learned directly from image-text pairs. We note that the input text description typically already contains detailed information on how to localize the target object, and we also observe that humans often follow a step-by-step comprehension process (\ie, progressively utilizing target-related attributes and relations as cues) to identify the target object. Hence, we propose a novel Progressive Comprehension Network (PCNet) to leverage target-related textual cues from the input description for progressively localizing the target object.Specifically, we first use a Large Language Model (LLM) to decompose the input text description into short phrases. These short phrases are taken as target-related cues and fed into a Conditional Referring Module (CRM) in multiple stages, to allow updating the referring text embedding and enhance the response map for target localization in a multi-stage manner.Based on the CRM, we then propose a Region-aware Shrinking (RaS) loss to constrain the visual localization to be conducted progressively in a coarse-to-fine manner across different stages.Finally, we introduce an Instance-aware Disambiguation (IaD) loss to suppress instance localization ambiguity by differentiating overlapping response maps generated by different referring texts on the same image. Extensive experiments show that our method outperforms SOTA methods on three common benchmarks.