Apollo Client with React Hooks allows developers to manually trigger client-side GraphQL queries using the `useLazyQuery` hook, enabling fetching data in response to events such as button clicks. This approach is useful when data needs to be fetched only when a specific condition is met or an event occurs, rather than on initial page load. By utilizing `useLazyQuery`, developers can control when and how their GraphQL queries are executed, providing more flexibility and customization options for their applications.