In my opinion, one of the best explanations for the 'Yippee-ki-yay' expression you heard in 'Die Hard' can be found at The greatest one-liner in movie history. - By Eric Lichtenfeld - Slate Magazine.
Here is a brief excerpt from that site: When terrorist-slash-exceptional thief Hans Gruber (Alan Rickman) taunts hero John McClane (Bruce Willis), "Who are you? Just another American who saw too many movies as a child?" and asks this "Mr. Cowboy" if he really thinks he stands a chance, McClane's answer—"Yippee-ki-yay, (expletive deleted)"—marks the moment that McClane, an everyman, assumes the mantle of America's archetypal heroes: Roy Rogers, John Wayne, Gunsmoke's Marshall Dillon, and others who have been so vital to American boyhood. Unlike the many action-movie one-liners that are rooted in the hero's narcissism, McClane's stems from our collective wish-fulfillment. He is not referring to himself, not suggesting an "I" or a "me" but an us. And considering the European Gruber's appreciation of fashion, finance, and the classics, McClane's comeback acquires an additional subtext: Our pop culture can beat up your high culture.
Basically, "Yippee ki-yay" is an old, American cowboy saying :"yippee" is a word used to express extreme joy, like "Whoopee!!" The Bruce Willis character uses the expression to summon the bravado that is symbolic of the legendary heroes of America's "Wild West" period.