To act boldly and put in the maximum effort (or else give
up altogether).
Possibly derived from the vocabulary of extreme sports such as freestyle downhill skiing or surfing in the 1980s, where the goal was to outdo other competitors by attempting bigger and more difficult challenges. The phrase was recorded earlier, however, suggesting that it may have already been established in areas such as horse racing (see www.barrypopik.com/index.php/new_york_city/entry/go_big_or_go_home).