Transylvania sounds like a vampire's address, but the name is plain geography. It means "beyond the forest," and the vampires came later.
Etymonline says the name is Medieval Latin, from trans, "beyond," and sylva, "forest." The name refers to the wooded mountains around the region. The earliest reference in the Wikipedia survey of its historical names is the Medieval Latin phrase terra ultra silvam, "land beyond the forest," in a document from 1075. A form Ultrasylvania is recorded in 1077, described as a direct translation of the Hungarian name.
That Hungarian name is the key. Erdély comes from Old Hungarian Erdeuelu, from erdő, "forest," plus the suffix elv, "beyond," used for border regions. The Latin Ultrasylvania is described as a direct translation of the Hungarian form, so the viewpoint is Hungarian too. The earliest recorded Hungarian form, Erdeuelu, appears in the 12th-century Gesta Hungarorum.
Other languages did not copy the forest idea. The German name Siebenbürgen means "seven fortresses." The most widely accepted explanation is that it refers to the seven main fortified towns of the Transylvanian Saxons, and it appears in a document of 1296. The Romanian name Ardeal was first recorded in 1432 as Ardeliu. The scholar Nicolae Drăganu argued in 1923 that it came from Hungarian Erdély, and the historian Ioan-Aurel Pop agrees.
Some languages do keep the forest. The Ukrainian name Zalissia also means "beyond the forest." Slavic names such as Polish Siedmiogród follow the German "seven" instead.
As for the vampire, Britannica notes that Bram Stoker's 1897 novel Dracula follows the count from his castle in Transylvania to England. That is the reason the name now carries a gothic tone. The forest, however, was there first.