I find it difficult to debug someone else's regexp. :-(
Before seeing @Ruger28's elegant solution, I would have done
matches = regexp(aMajorLine,'"[\w*,\\]*"','match')
matches = regexprep(matches,'"','');
Note that I first pull out with the double-quote characters, then strip them out in the second line. It is possible to do this in one regexp step instead, using "lookaround" assertions, but I find that less easy to understand.