Sachin,

This is a tough paper!  RNA splicing, stability and degradation are necessarily complex topics because these processes are so important in regulating gene function in cells.  To address your specific question, though, the metabolism of galactose vs. glucose is not the primary reason the researchers shifted their yeast from one medium to the other.  They are merely using galactose to conditionally express Rrp41 (for most of the paper).  This allele, GAL::rrp41, does not express the rrp41 gene under normal conditions (for instance, in glucose-only medium).  This is because instead of being driven by the normal rrp41 promoter, it is driven by the GAL10 promoter.   This promoter is normally inactive, but when the yeast cell senses galactose in the medium, it activates the GAL10 promoter.  This if analogous to the lac operon we all learned in high school.  Since the rrp41 gene is linked to the GAL10 promoter in these yeast cells, galactose activates transcription of rrp41.  So, what if they remove the galactose by changing the medium?  Rrp41 is no longer transcribed.  Because rrp41 is part of the exosome, they can either turn off the exosome by removing galactose, or turn on the exosome by providing galactose.  

I hope this helps in your studies!

Billy.