The invite comes in the form of an email from Electronic Arts which encourages players to "take part in an exclusive trial for an upcoming title on Microsoft's Xbox One." The email then sends Origin members to the application page on the official Titanfall website where they can login to apply. From there, you need only to login with your Origin credentials and your application will be submitted.
If selected to take part in the test, you will receive an email in "the coming days" with a unique code and further instructions on how to download and access the game on Xbox One. At this point, it's unclear what other qualifications one must have to be selected, but EA does warn that "availability to this trial is extremely limited."
Titanfall will be available for Xbox One, Xbox 360, and PC platforms on March 11, 2014 in North America. It'll be available in Europe two days later on March 13, 2014. Despite multiple rumors, the game is not a timed-exclusive and will be available only on the three aforementioned platforms for its lifetime.