The use of JSX in React allows developers to write HTML-like code within their JavaScript files, making it easier to construct and manage UI components. By encapsulating everything about a component into one file, including styles, JSX provides an elegant solution for working with Components. The example of the OpenSource modal component demonstrates how JSX can be used to create a reusable and customizable UI component, with all the necessary code and styling contained within the component itself. In contrast, Angular requires more boilerplate code for directives, which can make it less ideal for constructing UI components. The use of JSX in React also enables designers to style components without needing to write additional JavaScript code, making it a popular choice among developers and designers alike.