I'm gonna apologize up front if my answer seems vague or unhelpful... I'll trust that someone will fill in the blanks.
If you complete an interlink mission, you'll be rewarded with any number of 'bonuses' that will help you in your game play. When activated in your game, the bonuses can include speedy egg hatching, increased money earned in battle, and decreased prices at pokemarts... among many others.
One of these is the feature that enables your pokemon to gain experience at an increased rate... and by increased, I mean more than just the regular traded pokemon + lucky egg + elite 4 run/Audio farm.
I'm not clear on whether or not there is a specific mission that needs to be completed to gain this power, but my understanding is that once activated, it lasts only 3 minutes... and that this power + Audino farming gets the job done within that time.