From a study conducted by Salimpoor, Benovoy, et al.:
Music, an abstract stimulus, can arouse feelings of euphoria and craving, similar to tangible rewards that involve the striatal dopaminergic system.
These results indicate that intense pleasure in response to music can lead to dopamine release in the striatal system. Notably, the anticipation of an abstract reward can result in dopamine release in an anatomical pathway distinct from that associated with the peak pleasure itself.
This helps explain the basis for people crying when they're listening to music. As for why this tends to occur in a concert environment - this could be partly related to Salimpoor et al.'s suggestion that
composers and performers ... manipulate emotional arousal by violating expectations in certain ways or by delaying the predicted outcome (for example, by inserting unexpected notes or slowing tempo) before the resolution to heighten the motivation for completion.
One could argue that live performances tend to have more unexpected sequences like these (if you've been to a concert or listened to a live track, you notice that the live show might have a different, unexpected musical arrangement).
Additionally, a plausible explanation would be that when you have many people packed into the same space experiencing the music (as opposed to listening to it alone or in a small group), the emotional reactions discussed above could be amplified. There doesn't appear to be much research backing this up specifically, however.
Salimpoor, V. N., Benovoy, M., Larcher, K., Dagher, A., & Zatorre, R. J. (2011). Anatomically distinct dopamine release during anticipation and experience of peak emotion to music. Nature Neuroscience, 14(2), 257-262. doi:10.1038/nn.2726