What Linn has to say
Linn Vizard wrote an article called "Six Ways Stakeholders Use Service Blueprints — From a Real Project" that answers this exact question. Here is a short excerpt of it:
Created a shared view of the current state across different stakeholders
Diagnose the most resource intensive and painful parts of the process
Show evidence of the need for change and get buy in to try some changes
Train and on board new staff members
Outline the evolved process we were going to prototype and pilot
Document the newly implemented process
My two cents
Another use that I've witnessed for Service Blueprints that isn't in Linn's list is to imagine new service ideas or use the Service Blueprint as a prototyping tool.