What is meant by a royalty agreement and payments in conjunction with machinery that
is purchased under a royalty agreement that provides for payment of royalties based on units of production for the machines?

A royalty is where one company/individual is going to pay another company/individual for the use of something the second company owns or has some claim to. Normally, it's used for intellectual property. i.e. a book written, software, a movie, or even an idea. (e.g. I buy a book that the book store has purchased from the publisher. But the publisher will pay the writer of the book a royalty for each sold.) But I suppose it could be used for a plant asset also.

A royalty agreement is simply like any other agreement -- it's the agreement (hopefully on paper) of all the technicalities about how the property will be used and how the royalties will be paid.