The film Twelve Angry Men (this was also a play) comes to mind.
An influential experience, first and foremost, is going to change a person in an important way. In Twelve Angry Men, we see this happen most strikingly to Juror #3. Due to a prejudice against younger people, this juror makes the most vehement and emotionally biased case against the defendant, who is on trial for murdering his father.
Another juror, Juror #8, calmly and rationally points out to the jurors the holes in the prosecution’s case. Eventually he forces Juror #3 to see that his obstinance is based not on the reliability of the facts of the case, but on his own blind prejudice.
In order to have this influential experience, Juror #3 had to change. In order to change, he had to finally recognize his own motivations and faults. He was not able to do this on his own; it took the help of another person who recognized what was hindering him.
This play/movie was also influential with audiences because it placed a group of middle-class white men in a room together and forced them to deal with a racial issue. The defendant was hispanic (although not identified as such), and some of the jurors displayed prejudicial attitudes that were sure to mirror those of society in general. For an audience, seeing your own faults on stage or screen can be an influential experience, like holding a mirror up in front of a person.
To sum up, I'd say that integral elements of the influential experience include achieving some sort of new self-knowledge that leads to meaningful change in a person. It may be the efforts of another person that leads to the self-knowledge (like Juror #8), or our own introspection (thinking about the meaning behind 12 Angry Men).
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