Why is Jesus the only way? Because Jesus said that He was the only way. "I am the Way, and the Truth, and the Life; no one comes to the Father, but through Me," (John 14:6). Of course, just because someone says he is the only way, doesn't mean he is. But, this is why we need to look at Jesus' miracles as a proof that He was right about who He said He was. Jesus walked on water, calmed a storm with a command, raised people from the dead, and rose from the dead Himself. It is the fact of His incredible deeds.
Now, if Joe Schmoe on the street said that he was the only way to God, we'd look at him and say, "Yeah, right." But, if Joe calmed a storm with a command, raised someone from the dead, walked on water, etc., that would add a lot of credibility to his claim. After all, he is demonstrating extraordinary abilities consistent with his claim.
This is the case with Jesus. He made extraordinary claims and performed extraordinary deeds. Therefore, it is logical to conclude that what Jesus said was true--especially since He claimed to be God (John 8:24, 58; 10:30-33; 5:18).
Also, consider that no one else has done what Jesus has done. No one else has risen from the dead, calmed storms, raised others from the dead, and fulfilled numerous prophecies, etc. Though some may have claimed to be able to do one or two of these things, none have done all the things Christ has done as well as claim divinity. Since Christ has done all of these things and since He claims to be God in flesh, then it is logical to believe what He has said . . . that He is the only way.