You are actually not sending your login information to this site, when you press the Authenticate button below you are redirected to Geocaching.com where you will be asked to authenticate this site. When you press the Allow button, you will allow Project-GC to fetch some of the information bound to your Geocaching.com account. For example, your home coordinates. This data will be used in accordance with the Geocaching.com API License Agreement.
You can at any time you want, revoke Project-GC its rights by visiting Geocaching.com Authorizations.
The authentication used by the Geocaching Live API is based on a standard called oAuth2. This is the same technology that is used by Facebook and Twitter to allow other sites/applications to access accounts. Read more about oAuth2 here, or check out the Wikipedia page for easier reading.
By authenticating you will allow Project-GC to fetch information related to your Geocaching.com account in the background and for Project-GC to show detailed statistics about your finds and hides. You will also accept that you are not allowed to harm the site in any way. The use of bots, try to find exploits, spam the forum, or in other ways degrade the sites value or performance is included in those terms.
If you are entering an infinite loop while authenticating you should try logging out from Geocaching.com. Their session handling is having a bug for users that are not using Geocaching.com site in English.
If the above doesn't help you should try removing all cookies from Geocaching.com. If you do not know how to remove cookies you can check out this guide for example.
An alternative might be to revoke Project-GC its rights by visiting Geocaching.com Authorizations, and then try to authenticate again.