A PIECE OF THE ACTION means that someone wants to be involved in some popular or beneficial social activity.
For Example:
Due to the team’s success in recent games ticket sales have soared, and it seems everyone wants a piece of the action. – (Meaning lots of people want to see the team.)
Studying online is more convenient and often more interesting, so it’s not surprising that the number of students who want a piece of the action is growing rapidly. – (Meaning more and more students want to study online.)