The "Invalid License Key: Unable to connect to server" error is not caused directly by an invalid license key, but instead happens when MyBookTable is unable to connect to our servers to verify that it is indeed a valid license key. This means that there is something stopping your website from talking to our website, which is necessary in order to enable your new features.
First I would check that it is not a problem caused by another plugin or the theme on your website. Sometimes improperly coded themes or plugins can interfere with other plugins when they try to make outgoing connections. Please temporarily disable all other plugins and switch to a default WordPress theme, then try entering and saving the License Key again.
If that doesn't work, this is almost certainly a problem caused by your web hosting provider. Can you let us know what hosting provider you are using? You will probably need to contact them and tell them that your website is "unable to make outgoing HTTP connections to api.authormedia.com" and see if they can give you advice on how to fix it.