Update: Titanfall is once again free to play on Origin. New Titanfall players have 48 hours access which begins whenever the game is loaded up for the first time. The 48 hours then counts down in real time whether you are playing or not.
Original story: Titanfall is the first game to feature in EA's new Origin Game Time program, offering free and complete access to selected games for a limited time.
Users of Origin are able to play the full game for free as long as time remains on their Game Time clock. A maximum of 48 hours of Game Time is being offered for Titanfall this weekend.
The Origin Game Time clock begins as soon as you start the game for the first time, and counts down in realtime from this point on whether you're playing the game or not.
Origin Game Time games will appear on Origin's Free Games page when they’re available to download.
Source: EA Origin