How does react update part of the DOM?

A good place to get a better understanding of how react decides to re-render elements is the reconciliation documentation but I can summarize:

Every time render() is called react will create a new virtual DOM where the root node is the component whose render function is called. The render() function is called when either the state or the props of a component or any of its children change. The render() function destroys all of the old virtual DOM nodes starting from the root and creates a brand new virtual DOM.

In order to make sure the re-rendering of components is smooth and efficient React uses the Diffing Algorithm to reduce the time it takes to create a new tree to a time complexity of O(n), usually time complexity for copying trees is > O(n^2). The way it accomplishes this is by using the "key" attribute on each of the elements in the DOM. React knows that instead of creating each element from scratch it can check the "key" attribute on each node in the DOM. This is why you get a warning if you don't set the "key" attribute of each element, React uses the keys to vastly increase its rendering speed.

Execution of the render() method does not mean that react also renders the actual DOM. React keeps two copies of the virtual DOM i.e. the old virtual DOM and the re-rendered virtual DOM which gets created when the render() method is called.

The output of the render() method is a javascript object which is mapped to a DOM element. When a component is changed, we get a new react element. It then compares the new react element and its children in the re-rendered virtual DOM with the element and its children in the old virtual DOM. If any differences found, it then updates the real DOM only at the places where something has changed (e.g Text of the button has changed) and not update the entire real DOM. If no differences found, the real DOM is not touched.