Hi @yamasaki1ma, sorry for the late reply.
"Catch up bill run" is used to help customers go live. This is used for our internal team to migrate customer's data into Zuora. That's why you cannot find out any information about it from Knowledge Center, Community, and Google.
If customers have the old billing system, the subscription information needs to be migrated and Zuora needs continue to do billing for customers. The "Catch up bill run" helps continue to do billing. Without "Catch up bill run", Zuora internal team needs to create a bill run and generate lots of invoices and then do IIA to update invoice amount to 0.
Hope the above information is helpful.