Summary: River doesn't see why she shouldn't get to keep a few treasures.
River Song is an archaeologist by profession, and a thief by inclination. In her mind, they amount to the same thing since they both involve seeking out treasure: on the one hand, things long buried that belonged to vanished civilisations, and on the other, valuables that have been deliberately hidden away by their owners. Such things deserve to be seen and appreciated, not locked away in vaults, so she… liberates them.
Many people call her a criminal, but all she’s doing is redistributing wealth. Why should the rich and powerful have all the prettiest things? Doesn’t she deserve a share?