While implementing emp API in LWC to subscribe the platform event I  find out that there is small issue in sample code snipit given in documentation.
Issue is in handleSubscribe method in js below is the code given in documentation:

handleSubscribe() {
        // Callback invoked whenever a new event message is received
        const messageCallback = function(response) {
            console.log('New message received : ', JSON.stringify(response));
            // Response contains the payload of the new message received
        };

        // Invoke subscribe method of empApi. Pass reference to messageCallback
        subscribe(this.channelName, -1, messageCallback).then(response => {
            // Response contains the subscription information on successful subscribe call
            console.log('Successfully subscribed to : ', JSON.stringify(response.channel));
            this.subscription = response;
            this.toggleSubscribeButton(true);
        });
    }

 Correction is required in messageCallback method which should be as below:

const messageCallback = (response) => {
        console.log('New message received : ', JSON.stringify(response));
        this.payload = JSON.stringify(response);
        console.log('this.payload: ' + this.payload);
        // Response contains the payload of the new message received
    };

We have to use arrow function as it does not have its own scope.