The story begins at the end of the Dangeresque 3: The Criminal Projective when Strong Bad jumps off of the couch and hits the Trogdor arcade machine. The Trogdor arcade machine grows an arm and legs and comes to life. Strong Sad claims that Strong Bad damaged the 8-bit containment field and needs to order a replacement. Sadly, through an odd series of events, the containment field is broken, resulting in the world of videogames and the world of Free Country USA combining. Video game elements and characters begin invading the real world, and vice versa. In the ensuing chaos, Trogdor escapes his game and "burninates" Strong Badia. Strong Bad must defeat his fiery creation once and for all, by getting party members from other games to advance to the dungeon of the beast and restore balance between the two worlds.
Source: Wikipedia, "Strong Bad's Cool Game for Attractive People," available under the CC-BY-SA License.