They originally started as part of the Escapist, then after a dispute with them moved to Penny Arcade, then after PATV was shut down they just became a YouTube channel. Right around the point they moved to YouTube they got hired by the people who do the Total War series of games to do a video about history. Apparently the developers are fans of the show and discovered they have some leftover marketing money, so they just gave it to Extra Credits with no strings attached.
Extra Credits took the money and did a set of videos about the second Punic War. The response to that was apparently so good that they then did a video about how World War I got started (without covering any of the war itself,) and are currently in the middle of a series about the Warring States period of Japan.
Since this year is the 100th anniversary of the start of WWI let's list that one first.
Extra History - World War I: The Seminal Tragedy - Chapter 1: The Concert of Europe
Extra History - Rome: The Punic Wars - Chapter 1: The First Punic War
(Okay, it's mainly about the second Punic War, but they go over the first one in brief for the first episode just to give you a grounding for the second one.)
Extra History - Warring States Japan: Sengoku Jidai - Chapter 1: Battle of Okehazama