Hi Kane. Good question — I can see that it certainly sounds like an audacious claim when it’s put that way: “Mormons believe they are the only religion on earth with authority to baptize.”
The way that I would put it is this: When Jesus established his Church on the earth there was authority given to his apostles by the laying on of hands. His apostles had authority to baptize, and did so, but relatively soon thereafter, the apostles were killed and that authority was gone from the earth.
Some religions can trace this authority back pretty far, but even the Catholic Church has a very difficult time tracing it back to Jesus’ day, and acknowledge holes in the time period when the apostles were killed. We call this period of time the apostasy.
During the reformation, protestants like Luther, Zwingli and others recognized that while baptism was happening and that was good, the authority was missing. Baptism started to see some changes too (instead of creating expensive fonts or finding a river to baptism people by immersion (like Jesus was), they began trickling water and baptizing babies.
We (LDS) believe that at this time a young 14 year-old boy named Joseph Smith desperately wanted to join the correct religion with the proper authority. He studied the bible and ultimately took the question to God in a forrest near his house. We believe that Heavenly Father and Jesus appeared to him and told him that the authority had been taken from the earth, but that the time would soon come for it to be restored.
We believe that Joseph was later visited by a resurrected John the Baptist and was given the authority to baptize. Joseph then called 12 apostles and gave the authority to them. That authority has been passed down from person to person until now. I (for example) can trace my authority to baptize in about 13 steps to John the Baptist (through Joseph Smith).
So, we respect baptism in all religions and know that it is a wonderful a good practice — but the LDS Church does in fact believe that we have the authority to baptize.
You say that this religion has only been around for 200 years. We believe that Jesus Christ personally “restored” his original Church organization in modern times because the original Church authority was lost.
So we don’t believe that our Church is 200 years old. We believe it is the exact same Church that Christ formed when he walked on the earth.