A gasp is an audible (hear-able) noise emitted by an audience. It might be something like an "Oh!" or an "Ah!". Rapture is a pleasant feeling. .

"The Democratic Convention" would be the political meeting where the Democrat political party chooses their candidate for President. To "gasp" is to have trouble breathing for a moment, or to inhale suddenly.. You might gasp in surprise, or when seeing something very ugly— or very beautiful. So the film was shown at the Convention, with the reaction that people liked it so much they gasped.