Christians believe this because that's how Jesus described Himself. He claimed, "I am the way, the truth, and the life. No one can come to the Father except through me" (John 14:6). The phrase "no one" sounds all-inclusive, or all-exclusive, as the case may be. Jesus was either right or wrong when he said that. Those who are not His followers believe Jesus was wrong or misguided. Those who are His followers believe Jesus was right, and respectfully hold to His claim.
Many people believe that all religions are basically the same, and one way to God is as good as another. All you have to do is practice one of them and it works. Those who believe in Jesus respectfully disagree, for several reasons.
First, we disagree that all religions are basically the same. It would be clearer to say that all religions fit into two categories. One category offers a way to God that involves a system of rules and regulations. There are scores of these "effort-oriented" religions. All of these religions represent, at their core, a human-centred attempt to reach God or earn God's acceptance. Christianity fits into a different category altogether and it is unique. Christianity claims to be God's answer; God's revelation. It is not human-centred but God-centred. In fact, Christianity may include a lot of features that people identify as religion, but at its core Christianity is about a relationship with God. That relationship is God-given as a gift, not earned by effort or behaviour.
Christianity also insists that beliefs can't be "basically the same" if they contradict one another. If we insist that Jesus is the only way and other groups disagree, either they or we are wrong. If "basically the same" means that a miracle-less, divine-less, resurrection-less Jesus is basically the same as other religious leaders, then the statement may be true, but it isn't about Jesus any more.
The "basically the same" crowd sometimes ends up picturing a "way to God" that is easy, ignores the reality of sin and evil, and requires God to grade on the curve so that lots of morally mediocre people "reach God." The Bible presents a realistic and truthful picture of humanity when it declares that "everyone has sinned" (Romans 3:23), which means no one's effort is good enough to merit God's acceptance. When Jesus said he was the way, he wasn't referring to a system of rules; he was referring to himself as providing the way we could never find or achieve on our own.
The "basically the same" crowd want to be fair, but all they can offer is despair. They say they are open to all beliefs, when in fact they refuse to believe. Using another analogy, people say that God and his kingdom can be compared to a very big house. Like any house it has many doorways and access points. Every religion (in their view) offers an equally valid entrance. No matter which door you take, it gets you in the house. What this view fails to see is that believing in many doors in itself doesn't get anyone in the house. We can't go in a house through several doors at the same time. If we want to get inside, we have to pick a door. Until we choose a door, right or wrong, we remain outside, discussing the process. If we want to enter, we have to choose. People who say any door leads to God are avoiding choosing.
Christians say God's house is big enough for everyone, but it only has one door. Jesus said, "I am the gate. Those who come in through me will be saved" (John 10:9). Christianity doesn't stop people from trying other ways that claim to be "gates," but it insists that all other ways prove to be dead-ends except Jesus. The claim that Jesus is the only way to God isn't all that unusual (see John 14:6, quoted above). Most religious views claim they alone possess the key to access to God. Yet, they can't all be right, because among their teachings is the point that everyone else is wrong.
If what Jesus said is true, and there is only one way to have a relationship with God, then only one way can be right. The central proof behind Jesus' claim is his bodily resurrection from the dead (1 Corinthians 15:3-58). This is not a curious religious feature but a verifiable historical event. Other religious figures may have lived impressive lives, which Christians greatly respect. Other religions may include true moral and ethical teachings that Christians can admire. But when it comes to the issue of the human problem of sin from which we must be saved, Christians insist that only Jesus offers a free and living way. There are many ways to look for God; there's only one way to find Him - in Jesus Christ.
Believing there are many ways is a postponed choice.
Which way have you chosen, and why?
If you have a real problem with Jesus being the only way, then perhaps you don't really know Him yet.
If you haven't chosen yet, have you carefully examined the claims of Jesus?