This origin story for the Goths comes from Jordanes, a sixth-century historian of the Goths who wrote in Constantinople. Jordanes claimed to be of Gothic descent and wrote his "Getica" largely to justify the reign of Theoderic of the Ostrogoths in Italy. Jordanes suggested that the Goths, as a united group, originated on an island far to the north. The Swedish island of Gotland has been identified as possibly the island to which Jordanes was referring. However, it was a bit of a trope in Classical writing about ethnogenesis to claim that a particular group came from a far-off island, and there's no concrete material evidence to indicate that the Goths came from Gotland.

The first archaeological sign of the people who can possibly be labelled "Goths" is as part of the Wielbark culture in Poland (2nd and 3rd centuries AD). Wielbark culture was most likely an amalgam of various Germanic-speaking groups (and perhaps some Celtic- and proto-Slavic-speaking groups), of which the Goths were one small part.

The first unambiguous sign of the Goths is as part of the Chernyakhov culture north of the Black Sea (third and fourth centuries). Here, various Gothic kingdoms formed the dominant political element until the arrival of the Huns in the later fourth century. Some Gothic groups, like the Thervingi and Greuthungi, sought refuge in the empire. Other groups were conquered by the Huns.

The idea that the Goths migrated from the Wielbark area in northern Poland into the Chernyakhov area north of the Black Sea is contested. Peter Heather argues in favor of this hypothesis, but other historians, like Guy Halsall, argue that the Goths were products of the Roman frontier. In other words, no group identifying as "Goths" existed in the Wielbark cultural area. On the contrary, the "Goths" emerged north of the Black Sea as a result of interactions with the Roman Empire. The issue of ethnogenesis in this context is wrapped up in debates about group identity.

So to get around to finally answering your question, groups speaking a Gothic language probably migrated south from the Wielbark area in Poland to the area north of the Black Sea and formed kingdoms dominated by a Gothic-speaking warrior elite. Interactions with Rome then led to their identification as "Goths," a label that Goths themselves later adopted as a way of differentiating themselves from Romans once they'd moved into the Empire.

Sources: Jordanes, Getica; Heather, The Goths; Halsall, Barbarian Migrations and the Roman West

Edit: There's also the issue regarding the origins of the two Gothic groups that formed kingdoms in the empire: the Visigoths and Ostrogoths. I can answer that question as well if people are interested (it's not as simple as Wikipedia makes it out to be).