"SQL Injection" is subset of the an unverified/unsanitized user input vulnerability ("buffer overflows" are a different subset), and the idea is to convince the application to run SQL code that was not intended. If the application is creating SQL strings naively on the fly and then running them, it's straightforward to create some real surprises.
We'll note that this was a somewhat winding road with more than one wrong turn, and others with more experience will certainly have different -- and better -- approaches. But the fact that [we] were successful does suggest that we were not entirely misguided.
There have been other papers on SQL injection, including some that are much more detailed, but this one shows the rationale of discovery as much as the process of exploitation.