a.) "Here, we use a developmental approach to analyse molecular mechanisms in feather-branching morphogenesis": We are watching, at the molecular level, how different parts of a feather form as it grows.
b.) "We have used the replication competent avian sarcoma retrovirus" : We took a virus from birds -- a virus that was altered, but could still replicate: "...to deliver exogenous genes" : We used that virus to deliver genes created outside of the bird -- exo=outside, genous=created): "to regeneerating flight feather follicles of chickens."
So, we used a bird virus, put our controlled gene in the virus, used the virus to deliver the new genes to the replicating feather, and watched, at the molecular level, what happened when the controlled genes were expressed. Does that help :)