Assuming this Bronze Star was awarded for a specific deed of valor, the circumstances would have been outlined in the orders accompanying the award (the locale of the action was very often deleted for security reasons). Your Dad would have received a copy and another copy would have been placed in his service record. A copy of these orders is usually the only means of securing the info you're seeking. Embossed certificates were issued after the war but these were mostly decorative and usually indicated only the date and place of the action.
The Bronze Star may be awarded for individual heroism in NON-COMBAT as well as combat situations. When awarded for valor in actual combat, a small bronze letter "V" is authorized to be worn on the ribbon to so signify.
No reflection on Dear Ol' Dad but, through misinterpretation and/or misapplication of regulations, for years, from the late 1940's to the present, thousands of Bronze Stars have been issued solely on the basis of the individual having received the Combat Infantryman Badge (CIB). You can determine this by requesting a statement of service from the NPRC in St. Louis. The Bronze Star will be listed with the caveat, "(based on Combat Infantryman Badge)".