What is {this.props.children} and when you should use it?

I assume you're seeing this in a React component's render method, like this (edit: your edited question does indeed show that):

class Example extends React.Component {
  render() {
    return <div>
      <div>Children ({this.props.children.length}):</div>
      {this.props.children}
    </div>;
  }
}

class Widget extends React.Component {
  render() {
    return <div>
      <div>First <code>Example</code>:</div>
      <Example>
        <div>1</div>
        <div>2</div>
        <div>3</div>
      </Example>
      <div>Second <code>Example</code> with different children:</div>
      <Example>
        <div>A</div>
        <div>B</div>
      </Example>
    </div>;
  }
}

ReactDOM.render(
  <Widget/>,
  document.getElementById("root")
);

<div id="root"></div>
<script src="https://cdnjs.cloudflare.com/ajax/libs/react/15.1.0/react.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/react/15.1.0/react-dom.min.js"></script>

children is a special property of React components which contains any child elements defined within the component, e.g. the divs inside Example above. {this.props.children} includes those children in the rendered result.

...what are the situations to use the same

You'd do it when you want to include the child elements in the rendered output directly, unchanged; and not if you didn't.

What even is ‘children’?

The React docs say that you can use props.children on components that represent ‘generic boxes’ and that don’t know their children ahead of time. For me, that didn’t really clear things up. I’m sure for some, that definition makes perfect sense but it didn’t for me.

My simple explanation of what this.props.children does is that it is used to display whatever you include between the opening and closing tags when invoking a component.

A simple example:

Here’s an example of a stateless function that is used to create a component. Again, since this is a function, there is no this keyword so just use props.children

const Picture = (props) => {
  return (
    <div>
      <img src={props.src}/>
      {props.children}
    </div>
  )
}

This component contains an <img> that is receiving some props and then it is displaying {props.children}.

Whenever this component is invoked {props.children} will also be displayed and this is just a reference to what is between the opening and closing tags of the component.

//App.js
render () {
  return (
    <div className='container'>
      <Picture key={picture.id} src={picture.src}>
          //what is placed here is passed as props.children  
      </Picture>
    </div>
  )
}

Instead of invoking the component with a self-closing tag <Picture /> if you invoke it will full opening and closing tags <Picture> </Picture> you can then place more code between it.

This de-couples the <Picture> component from its content and makes it more reusable.

Reference: A quick intro to React’s props.children