"Red letters" and "black letters" in reference to the Bible are, admittedly, a bit archaic. I'm sure some Bibles still do this, but for some time Bibles were printed with the words of Jesus set in red ink. So, in theory at least, "red letter Christians" neglect the teachings of, and stories about, the prophets, the patriarchs, the apostles, the early church, the Israelites--everyone but Jesus. In fact, even the acts of Jesus are seemingly set aside by "red letter Christians," leaving only the words that came out of his mouth.
That would be a thin faith indeed. Thomas Jefferson cut all the miracles out of his Bible, and what he had left would still be hundreds of pages longer than what a red letter Christian would keep. So yeah, if that's what red letter Christians are all about, they start to sound like a cult.
A more charitable take on red letter Christians might be to say that they elevate Jesus' words over the rest of the Bible. This still raises the hackles of critics, however, who champion the equal weight of the whole canon of Scripture. So, if red letter Christians are creating a hierarchy of authority within the Bible, that becomes a problem too.
Fortunately, that's not what red letter Christians are doing. Tony Campolo wrote his take on the controversy here, where he attempts to put his movement in the context of other Christian movements:
Much in the same way that certain churches identify with Micah 6:8 or John 3:16 our movement identifies with the specific words of Christ with regards to action and deed. . . . Whether Christ is referring back to Leviticus or Isaiah, the Red Letters vividly show the connection between Christ’s words and the whole of Scripture. As Jesus himself says: “Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them.”
Finally, the purpose of Red Letter Christians is not to establish a new theology but to assist and encourage followers of Jesus to live out the lifestyle prescribed by Christ.
Jamie Arpin-Ricci also helpfully weighs in from his blog:
First, “red letter Christian” is not the arching term with which I define my whole faith, any more than being called a husband makes me any less a son or brother or father. So while it reflects an important commitment, it should not be understood as an affirmation over and against other aspects of my wider faith.
Second, the concept of Red Letter Christians was obviously inspired by the Red Letter Edition of the Bible, invented by editor Louis Klopsch, in which the words of Jesus appear in red ink. Klopsch made it abundantly clear that this innovation was not intended to elevated one part of Scripture over any other, but rather to actually allow us to focus on the fullness of what Jesus said, seeing how powerfully it reflected the truth and fulfillment of the rest of Scripture.
Jamie goes on to emphasize the real meaning behind the movement: to showcase the idea of Jesus as a provocateur. We hear "the teachings of Jesus" and too easily slip into the sanitized, suburbanized interpretations of those teachings that constitute the mainstream teachings of the contemporary church. In reality, Jesus was as much provoker as teacher, confronting established understandings of biblical texts, challenging accepted forms of piety and worship, decrying the injustice of the culture of his day. Red letter Christians refuse to hide from the uncomfortable truths coming out of Jesus' mouth. We don't necessarily handle those truths consistently better than anybody else, but we're committed to keeping them in front of us. I'll give Jamie the last word:
That some might take such a commitment too far is a fair concern. However, such concern can become just as much a hindrance when it keeps us from facing the prophetically painful questions that Jesus’ words & example confront us with.