Several things come to mind. The firstborn gets more individual attention from parents. Parents tend to really get in tune with the firstborn, everything they do, say, every gurgle. That attention creates a certain self-assurance.
The next child comes along and the firstborn now has to "compete" to keep that attention they have grown accustomed to--sometimes by bullying the second child so that child 1 gets his/her way.
Now add a third child and the parents start to depend on the firstborn to help out depending on the child's age. they can get diapers, alert the mom to the baby crying and so on--thus giving child one a feeling of responsibility.