Why is PBS added in the last step of the Alizarin Red staining protocol (“Osteogenic differentiation and analysis of MSC”)? Should the PBS be aspirated before analysis?

The PBS buffering enhances the Alizarin Red staining (precipitation of the dye) and makes it more intense.
Leave the PBS on the cells after staining/washing and analyze the sample immediately, as the dye may bleed upon prolonged storage without embedding.