class App extends React.Component {
  state = {
    show: true
  };

  showhide = () => {
    this.setState({ show: !this.state.show });
  };

  render() {
    return (
      <div className="App">
        {this.state.show && 
          <img src={logo} className="App-logo" alt="logo" />
        }
        <a onClick={this.showhide}>Show Hide</a>
      </div>
    );
  }
}

var Search= React.createClass({
 getInitialState: () => { showResults: false },
 onClick: () => this.setState({ showResults: true }),
 render: function () {
   const { showResults } = this.state;
   return (
     <div className="date-range">
       <input type="submit" value="Search" onClick={this.handleClick} />
       {showResults && <Results />}
     </div>
   );
 }
});

var Results = React.createClass({
    render: function () {
        return (
            <div id="results" className="search-results">
                Some Results
            </div>
        );
    }
});

React.renderComponent(<Search /> , document.body);

Simple hide/show example with React Hooks: (srry about no fiddle)

const Example = () => {

  const [show, setShow] = useState(false);

  return (
    <div>
      <p>Show state: {show}</p>
      {show == true ? (
        <p>You can see me!</p>
      ) : null}
      <button onClick={() => setShow(!show)}>
    </div>
  );

};

export default Example;

Use this lean and short syntax:

{ this.state.show && <MyCustomComponent /> }

this example shows how you can switch between components by using a toggle which switches after every 1sec

import React ,{Fragment,Component} from "react";
import ReactDOM from "react-dom";

import "./styles.css";

const Component1 = () =>(
  <div>
    <img 
src="https://i.pinimg.com/originals/58/df/1d/58df1d8bf372ade04781b8d4b2549ee6.jpg" />
   </div>
)

const Component2 = () => {
  return (
    <div>
       <img 
src="http://www.chinabuddhismencyclopedia.com/en/images/thumb/2/2e/12ccse.jpg/250px- 
12ccse.jpg" />
  </div>
   )

 }

 class App extends Component {
   constructor(props) {
     super(props);
    this.state = { 
      toggleFlag:false
     }
   }
   timer=()=> {
    this.setState({toggleFlag:!this.state.toggleFlag})
  }
  componentDidMount() {
    setInterval(this.timer, 1000);
   }
  render(){
     let { toggleFlag} = this.state
    return (
      <Fragment>
        {toggleFlag ? <Component1 /> : <Component2 />}
       </Fragment>
    )
  }
}


const rootElement = document.getElementById("root");
ReactDOM.render(<App />, rootElement);

This can also be achieved like this (very easy way)

 class app extends Component {
   state = {
     show: false
   };
 toggle= () => {
   var res = this.state.show;
   this.setState({ show: !res });
 };
render() {
  return(
   <button onClick={ this.toggle }> Toggle </button>
  {
    this.state.show ? (<div> HELLO </div>) : null
  }
   );
     }

If you use bootstrap 4, you can hide element that way

className={this.state.hideElement ? "invisible" : "visible"}

In some cases higher order component might be useful:

Create higher order component:

export var HidableComponent = (ComposedComponent) => class extends React.Component {
    render() {
        if ((this.props.shouldHide!=null && this.props.shouldHide()) || this.props.hidden)
            return null;
        return <ComposedComponent {...this.props}  />;
    }
};

Extend your own component:

export const MyComp= HidableComponent(MyCompBasic);

Then you can use it like this:

<MyComp hidden={true} ... />
<MyComp shouldHide={this.props.useSomeFunctionHere} ... />

This reduces a bit boilerplate and enforces sticking to naming conventions, however please be aware of that MyComp will still be instantiated - the way to omit is was mentioned earlier:

{ !hidden && <MyComp ... /> }

   class FormPage extends React.Component{
      constructor(props){
           super(props);
           this.state = {
             hidediv: false
           }
      }

     handleClick = (){
       this.setState({
          hidediv: true
        });
      }

      render(){
        return(
        <div>
          <div className="date-range" hidden = {this.state.hidediv}>
               <input type="submit" value="Search" onClick={this.handleClick} />
          </div>
          <div id="results" className="search-results" hidden = {!this.state.hidediv}>
                        Some Results
           </div>
        </div>
        );
      }
  }

Best practice is below according to the documentation:

{this.state.showFooter && <Footer />}

Render the element only when the state is valid.

I start with this statement from the React team:

In React, you can create distinct components that encapsulate behaviour you need. Then, you can render only some of them, depending on the state of your application.

Conditional rendering in React works the same way conditions work in JavaScript. Use JavaScript operators like if or the conditional operator to create elements representing the current state, and let React update the UI to match them.

You basically need to show the component when the button gets clicked, you can do it two ways, using pure React or using CSS, using pure React way, you can do something like below code in your case, so in the first run, results are not showing as hideResults is true, but by clicking on the button, state gonna change and hideResults is false and the component get rendered again with the new value conditions, this is very common use of changing component view in React...

var Search = React.createClass({
  getInitialState: function() {
    return { hideResults: true };
  },

  handleClick: function() {
    this.setState({ hideResults: false });
  },

  render: function() {
    return (
      <div>
        <input type="submit" value="Search" onClick={this.handleClick} />
        { !this.state.hideResults && <Results /> }
      </div> );
  }

});

var Results = React.createClass({
  render: function() {
    return (
    <div id="results" className="search-results">
      Some Results
    </div>);
   }
});

ReactDOM.render(<Search />, document.body);

If you want to do further study in conditional rendering in React, have a look here.

with the newest version react 0.11 you can also just return null to have no content rendered.

Rendering to null