I believe all this is referring to is using the slow application of load in a dynamic analysis in order to get essentially static results. My guess is there is something about the non-linear contact elements that make the solution from a "dynamic analysis" more likely to converge than when doing an true static analysis.
For what it's worth, I have used "quasi-static" analyses when I wanted to verify that our dynamic analysis is working correctly. That means I applied the load slowly, and compared it to the static analysis results that I got. Granted, my situation was a lot easier than what they're talking about in the document, because mine was merely a linear analysis.