I think I found your problem. In your handleClick method, you change the state to false to hide your list. When you press the button again to display the list, all they do is set the status to false again. Flip the state as follows:
handleClick() {
this.setState({ linksHidden: !this.state.linksHidden });
}
: . , , setState , , .
handleClick() {
this.setState(state => ({ ...state, linksHidden: !state.linksHidden }));
}
, CSS. . , react-motion. , .