Why or how did Valentine's Day begin?

Answer 1

Every February in ancient Rome, teenage boys and girls celebrated fertility and choose a mate. The god of flocks was named Lupercus and this was part of the celebration. When Christianity became the religion of the Roman Empire the holiday was abolished. A pope named Gelasius replaced Lupercus with Saint Valentine. As the holiday evolved it became Valentine's Day.

Another answer:

In the 3rd century AD was a Christian martyr, a priest named Valentine, who would later be named a saint (by the Catholic Church). The love/romance part comes from a legend that Roman Emperor Claudius II, ordered the young men to remain single because he believed that married men did not make good soldiers. The priest Valentine, however, secretly performed marriage ceremonies for the young men.
St. Valentine was, against the edict of the Caesar, performing marriages for the Roman soldiers who wanted the women they loved as their wives, as they were Christians, and wanted to legitimize their love. He did this under threat of execution, and married the couples in secret. He was eventually discovered, and since he had disobeyed an Imperial order, was executed. The evening before his impending death, he wrote a note to his converts telling them to keep the faith, and signed it "Your Valentine". He is the patron Saint of Love because he was martyred it's name.