react calling a function in a child component

The easiest way to achieve this is through using refs(See documentation).

var Parent = React.createClass ({

   triggerChildDisplay: function() {
       this.refs.child.display();
   },

   render() {
       <Button onClick={this.triggerChildDisplay} />
   } 
})


var Child = React.createClass ({
    getInitialState () {
        return {
            display: true
        };
    },


    display: function(){
        this.setState({
             display: !this.state.display
        })
    },

    render() {
     {this.state.display}
    } 
})

I basically copied in your example, but do note that in your Parent, I don't see you render a Child component, but typically, you would, and on that Child you would give it a ref, like <Child ref="child" />.