A prior engagement is a plan to do something that you made before an offer of new plans came up. It's similar to prior commitment, previous commitment, or having other plans.
Prior means preceding in order; earlier.
Engagement is an agreement to be somewhere at a chosen time.
Prior engagement is also the name of a three piece band based in the Hampshire County, England. They play the blues based with some blues rock/blues jazz. They play the music of Rory Gallagher, Thin Lizzy, and Dire Straits among others.
Check out 7 Tips for Learning English Vocabulary, shared by English Club.