Why are cost per output and cost per outcome calculations valuable to a human service agency?

By definition , one gets into a business to make a profit.
It goes without saying that to sell a service
At a profit one needs necessarily to know
What the cost of that service is.

What does it mean by cost of outcome and cost per output?

You have a field day here.
This depends on your definitions:-
1) Outcome.
2) Output.

What comes to mind for outcome is a big
Building project construction.

What output is can be just a box of matches
Or 1 Million boxes of matches.

There are many examples of both.
You just have to think out of the box
Of THE PERIOD CONVENTION.