Much of the text of the constitution and bill of rights seems to based on context that is now lost. The argument over the right to bear arms is always one of context. I.e. did the authors mean we need a well-formed militia, or did they mean to say everyone may carry without government restriction? The third amendment is another which has a context that seems to no longer apply.
What is the context of the right to petition? Was it expected that any person should be able to appear before congress with grievances? The Obama administration has made some attempt to meet this need with the petition system on whitehouse.gov, but that doesn't seem to make a difference. Was the government meant to respond to such petitions, or only hear them? How was this handled in the past and why was it important to the authors of the Bill of Rights?
This is an extremely broad question, so I'm going to answer in an abbreviated fashion. Feel free to ask follow-up questions if needed.
First, "petition" as understood by the Framers of the First Amendment had a specific meaning that was informed by centuries of practice in England, the colonies, and the Confederation. "By the seventeenth century, then, what was a petition? A petition was a communication that, 1) had to be addressed to an authority such as the King, 2) had to state a grievance, and, 3) had to pray for relief." Gregory A. Mark, The Vestigal Constitution: The History and Significance of the Right to Petition, 66 Ford. L. Rev. 2153, 2173 (1998). Petitions were not in-person appearances before Congress, but were rather legal documents that were addressed to Congress and requested specific relief.
The original understanding of the Right to Petition quickly fell by the wayside, as Congress was overwhelmed by petitions, particularly petitions regarding the abolition of slavery, which resulted in an infamous "gag rule." However, even putting aside that issue, the sheer volume of petitions meant that the original understanding of the amendment was never realized: "With such experience as a guide, the early congresses naturally 'attempted to pass favorably or unfavorably on every petition.' Such consideration did not last. Historians and other scholars have thus concluded that petitioning met its demise soon after (or at least within a few decades after) it was secured in the First Amendment. The volume of petitions Congress received may have contributed to the transformation of the right. Indeed, the traditional explanation is that volume debased the right, precipitating its demise." Id.
The Right to Petition was important to the Anti-Federalists because they felt that the Constitution was too anti-democratic. Petitions would be missives “of a free people [to] their servants.” While in presenting petitions “[p]ropriety requires that the people should approach their representatives with a becoming humility,” the “governors . . . as their servants . . . are . . . bound to observe decency towards them, and to act according to their instructions and agreeably to conscience.” Philadelphiensis, No. 5, reprinted in 3 The Complete Anti-Federalist 116 (Herbert J. Storing ed., 1981).