React.js: How to append a component on click?

Don't use jQuery to manipulate the DOM when you're using React. React components should render a representation of what they should look like given a certain state; what DOM that translates to is taken care of by React itself.

What you want to do is store the "state which determines what gets rendered" higher up the chain, and pass it down. If you are rendering n children, that state should be "owned" by whatever contains your component. eg:

class AppComponent extends React.Component {
  state = {
    numChildren: 0
  }

  render () {
    const children = [];

    for (var i = 0; i < this.state.numChildren; i += 1) {
      children.push(<ChildComponent key={i} number={i} />);
    };

    return (
      <ParentComponent addChild={this.onAddChild}>
        {children}
      </ParentComponent>
    );
  }

  onAddChild = () => {
    this.setState({
      numChildren: this.state.numChildren + 1
    });
  }
}

const ParentComponent = props => (
  <div className="card calculator">
    <p><a href="#" onClick={props.addChild}>Add Another Child Component</a></p>
    <div id="children-pane">
      {props.children}
    </div>
  </div>
);

const ChildComponent = props => <div>{"I am child " + props.number}</div>;

As @Alex McMillan mentioned, use state to dictate what should be rendered in the dom.

In the example below I have an input field and I want to add a second one when the user clicks the button, the onClick event handler calls handleAddSecondInput( ) which changes inputLinkClicked to true. I am using a ternary operator to check for the truthy state, which renders the second input field

class HealthConditions extends React.Component {
  constructor(props) {
    super(props);


    this.state = {
      inputLinkClicked: false
    }
  }

  handleAddSecondInput() {
    this.setState({
      inputLinkClicked: true
    })
  }


  render() {
    return(
      <main id="wrapper" className="" data-reset-cookie-tab>
        <div id="content" role="main">
          <div className="inner-block">

            <H1Heading title="Tell us about any disabilities, illnesses or ongoing conditions"/>

            <InputField label="Name of condition"
              InputType="text"
              InputId="id-condition"
              InputName="condition"
            />

            {
              this.state.inputLinkClicked?

              <InputField label=""
                InputType="text"
                InputId="id-condition2"
                InputName="condition2"
              />

              :

              <div></div>
            }

            <button
              type="button"
              className="make-button-link"
              data-add-button=""
              href="#"
              onClick={this.handleAddSecondInput}
            >
              Add a condition
            </button>

            <FormButton buttonLabel="Next"
              handleSubmit={this.handleSubmit}
              linkto={
                this.state.illnessOrDisability === 'true' ?
                "/404"
                :
                "/add-your-details"
              }
            />

            <BackLink backLink="/add-your-details" />

          </div>
         </div>
      </main>
    );
  }
}