I understand their ability to block legislation was a powerful one, but I'm having trouble understanding why the position became so politically important/powerful as time went on, and especially in the late republic.
They not only had the ability to block legislation, but to convene comitia plebis and pass laws that only affected the plebes.
As the empire aged the decrees of the comitia plebis changed the law for all Roman citizens, making the power of the Tribune of the Plebs much greater.
Later still, the Tribune of the Plebs could propose legislation to the Senate.
Someone with more than the passing knowledge I have will no doubt wander by soon and give a more in-depth answer, but this will get you started I hope.