Typically done when a person uses slightly relevant knowledge to demonstrate their superiority over someone. Usually intended to belittle a person and make them feel insignificant or "show them up"
When Tina Weaver shows up and attempts to wax corporate-speak in a vein attempt to sound like she understands business, that is big leaguing. In this case she has little to no business knowledge but to the unsuspecting greeter, she appears competent and thus successfully big leagues
by Lance Rice December 12, 2008