First off, some bird species are most active at night, owls for example. Members of the goatsucker family are also night timers - google for nighthawk - a very common bird.
The site below has lots of bird vocalizations. One species can make different vocalizations.
http://www.allaboutbirds.org/guide/Savannah_Sparrow/sounds/ac
Since birds may have a variety of calls: warning, mating, territory, and some species have many more types of calls than others (example: birds in the crow family, Corvidae.)
what is happening is that there is an increase of different vocalizations across species.
Kind of like checking on where you wife is. And worrying about the owl coming back to roost nearby
So during dawn and dusk, you may hear bird calls that relate to locating mates nearby, re-bonding, and letting others of the same species know of the caller's presence, plus additional frequency of warning calls.
Warning calls - Dawn and dusk is a very active time for predators, in part because both shifts of predators are active , diurnal and nocturnal. This causes increased numbers of warning calls. Interestingly, most species of woodland birds know the warning vocalizations of other species of birds, a sort of a mutual defense network. So when one species starts squawking about bad guys nearby, others will pick up and add warning in ther own "language".
You will also notice a decline in dusk and dawn bird calls during winter, when a lot of species may not mate or may have migrated South.