What is the purpose of spreading props in React Higher Order Components?

The answer is directly explained in the docs:

Convention: Pass Unrelated Props Through to the Wrapped Component HOCs add features to a component. They shouldn’t drastically alter its contract. It’s expected that the component returned from a HOC has a similar interface to the wrapped component.

HOCs should pass through props that are unrelated to its specific concern. Most HOCs contain a render method that looks something like this:

To understand this you should know what {...this.props} does. In your case

const EnhanceComponent = BaseComponent => {
    return class EnhancedComponent extends Component {
        state = {
            name: 'You have been enhanced'
        }
        render() {
           return ( 
           <BaseComponent {...this.props} {...this.state} />   
        )
       }
    }
};

export default EnhanceComponent;

EnhanceComponent HOC does a simple operation of adding a state name to the component currently being rendered, so essentially when you use this HOC, you should be able to pass the props required by your original component directly to it rather than consuming them in the HOC, which is what {...this.props} spread syntax is for. You can read this answer for more details on how ... works

Consider the case of a simple component which is used like

<MyComponent className='wrapper-container' onClick={this.handleClick} />

and defined as

class MyComponent extends React.Component {
      render() {
         const { className, onClick} = this.props;

         ...
      }
   }

Now if you use an HOC over this component like

const EnhancedMyComponent = EnhanceComponent(MyComponent);

You would render it like

<EnhancedMyComponent className='wrapper-container' onClick={this.handleClick} />

and now if you don't write {...this.props} in your HOC, then the MyComponent will no longer have className and onClick as props