Approbation is an official, important-sounding, and somewhat old-fashioned word for approval or praise. A princess, for example, might only consider marrying a prince that is met with her father's, the King's, approbation.
How is it possible that approbation means approval when probation is a form of being in trouble in school? Remember that probation is a testing period, to see if you can be good. Approbation means it's all good. Or you can remember this rhyme: "Filled with approbation, the audience gave a standing ovation."