The origin of agarrón is straightforward. When someone punches several times some other person, the former agarra a chingadazos the latter. Agarrar a chingadazos means "to grab with punches". The same happens when someone agarra a patadas (kicks) or cachetadas (slaps on the face) someone else.
Agarrón is used by sports commentators in the media to speak of an interesting and rather balanced match. Typcially, games between Chivas and América (the two most popular soccer teams in Mexico) are agarrones, and from time to time some players se dan un agarrón.