A different example for this use of of the preposition to:
he danced to the music
In this sense, the dance is performed to fit to the music. This aspect is less notable with applause because there is no rhythm, but other facets, are relevant nevertheless, e.g. ambiance or dynamic. The metaphorical aspect of moving towards the "music" is less obvious when leaving the stage, but it still ascribes a positive aspect, ie: more than "under applause"; not as strong as "in spite of"; and in stark contrast to "because of".
Edit to answer the second part of the question:
"prolonged" is a participle (edit2: well, not quite, depends who you ask, it's an adjective, alright, some would say an adjectival participle is not a real participle, if they'd call this adjectival participle at all).
"..., the audience applauding" or "while the audience was applauding" would be participle constructions to use the continuous tenses, which would fit better to highlight the progressive nature of ... the progress. But I can't advise which to prefer.