The react-apollo project aims to simplify the integration of GraphQL with React by providing a higher-order component (HOC) that wraps around the `graphql` function from Apollo Client. This HOC allows developers to describe their data requirements and convert the results into the shape required by their components, making it easier to connect their app to server-side data. The project solves common difficulties related to combining GraphQL and React, such as colocation of data, meaningful props, and easy configuration. It also enables server-side rendering, testing, and support for React Native. By using react-apollo, developers can take advantage of the benefits of GraphQL while maintaining a seamless integration with React.