Query understanding is a crucial component of Information Retrieval (IR), aimed at identifying the underlying search intent of textual queries. However, most existing approaches oversimplify this task into query classification or clustering, which fails to fully capture the nuanced intent behind the query. In this paper, we address the task of query intent generation: to automatically generate detailed and precise intent descriptions for search queries using relevant and irrelevant documents given a query. These intent descriptions can help users understand why the search engine considered the top-ranked documents relevant, and provide more transparency to the retrieval process. We propose a dual-space model that uses semantic relevance and irrelevance information in the returned documents to explain the understanding of the query intent. Specifically, in the encoding process, we project, separate, and distinguish relevant and irrelevant documents in the representation space. Then, we introduce a semantic decoupling model in the novel disentangling space, where the semantics of irrelevant information are removed from the relevant space, ensuring that only the essential and relevant intent is captured. This process refines the understanding of the query and provides more accurate explanations for the search results. Experiments on benchmark data demonstrate that our methods produce high-quality query intent descriptions, outperforming existing methods for this task, as well as state-of-the-art query-based summarization methods. A token-level visualization of attention scores reveals that our model effectively reduces the focus on irrelevant intent topics. Our findings open up promising research and application directions for query intent generation, particularly in exploratory search.