From Vault of the Gameverse:
The story of Titanfall is told through narration in the match Lobby, while in-game playing a match, and when a match comes to an end. It is very understandable that players are not following it. Titanfall is one of the most awesome and hectic multiplayer shooters ever. Probably the most. Who is paying attention to narration and little tiny in-game videos while they’re running around trying to dominate other players? But that is how the campaign is played out.
When it comes to multiplayer, there is nothing else like it.
Unfortunately no one can be told what Titanfall is. You have to play it for yourself.