This is an excellent answer by u/GreatThunderOwl.
The short and simple answer as to why death metal, and extreme metal in general, became the dominant form of metal in Sweden, Norway, and Finland, is that it was the continuation of a trend in metal going back to the early 80s. That is, nearly every metal scene was moving toward more and more extreme versions of metal, and that extreme metal genres began to dominate the underground metal sound globally starting in the late 80s (and continue to dominate the metal scene to this day). The Swedish and Norwegian scenes became particularly popular in the underground scene in the 90s, with the Gothenburg scene becoming the most popular scene of the late 90s/early 2000s (In Flames, At the Gates, Dark Tranquillity, etc). The Finish scene didn't really rise to prominence until the early/mid 2000s (starting with Finntroll's 2001 album Jaktens tid), with the exception of Amorphis who were one of the early pioneers in the melodic death metal sub genre.
A longer answer:
Heavy metal in the 80s was an arms race in extremes. Bands were attempting to play as fast as possible, as technical as possible, as theatrically as possible. Bands were competing to be the loudest, to have the most riffs in a single track, the most impressive riffs, the most soaring vocal, the most extreme visual look, to be the most ominous, etc. Out of that arms race emerged the extreme metal genres. In the Anglosphere, this was mostly in the form of death metal (centered on Florida, New York and in England in the form of grind). In continental Europe, this was mostly in the form of black metal, emerging out of the "Teutonic thrash" of Germany, Switzerland, Austria and the Czech Republic, mixed with the influence of UK bands like Venom and Motorhead.
The Norwegian and Swedish metal scenes had their own personal arms race in the 90s. The two metal scenes competed with one another and played pranks on the other and even fought with one another. The competition and animosity between the two scenes played a major role in moving them in separate directions. The Swedish scene embraced the Anglo death metal, while the Norwegian scene embraced the continental black metal. As the two scenes competed with one another, their national sound became more and more distinct, with the Swedish sound becoming famous for it's extensive use of melody and hooks, while the Norwegian scene moved in the opposite direction and became a cacophonous nearly impenetrable wall of sound.
The extensive use of melody and hooks within the Swedish death metal scene meant that it was very catchy, and quickly gained a global audience. The Norwegian scene became famous more for it's actions than it's sound. Norway's Black Circle was lead by a right-wing terrorist who wanted to exclude Jewish and Christian influences from Norway, and were responsible for the arson attempts on some 50 churches in Norway (some successful), as well as a few murders within the scene. The violence and infamy of the Norwegian scene was heavily romanticized in underground metal circles, which lead to it's popularity among the underground, but it didn't begin to match the global success of the Swedish death metal scene until after the 90s with the rise of more accessible progressive black metal and symphonic black metal sub genres.
The Finish scene didn't really rise to prominence until fairly recently (Moonsorrow's first US show was in 2006 at the Heathen Crusade metal festival, but they didn't do a US tour until 2009). The Finish scene is known mostly for it's folk and viking metal, and like the Swedish scene, owes a major part of it's success to it's catchiness. The Finish scene also rode the rise of viking metal, as the genre began to become one of the most popular forms of metal among the underground as Gothenburg metal lost popularity.
Sources: Lords of Chaos: The Bloody Rise of the Satanic Metal Underground -- Michael Moynihan and Didrik Søderlind
Sound of the Beast: The Complete Headbanging History of Heavy Metal -- Ian Christie
Choosing Death: The Improbable History of Death Metal & Grindcore -- Albert Mudrian