That's like asking "What lead North Americans from Revolutionary War Militia Culture into organizations like NAFTA?"

There was no Viking people, or culture. There were Danes, Norwegians, Swedes (proper), Geats, Gutes, further subdivided into groups like Jutes, Trønder, West-Geats, Rosbyggjar, et cetera. It is by those groups they self-identified and are identified in Scandinavian accounts. They had a common language - Old Norse - (and the languages are still largely mutually intelligible), which diverged in the Viking Age into East and West Norse. And in the time period you're addressing, East Norse diverged further into Old Danish and Old Swedish.

A 'viking' in their terminology was someone out of the minority group who went off on trips. Whether that was to trade peacefully or raid, did not matter - and lots of them were trade trips, particularly in the east. But the 'viking' image is not only defined by foreign accounts, but by west-European ones as well. Anyway, nor was being a 'viking' a a full time occupation. In modern English usage it refers to those who went off between 793 and 1066. While the raiders on the Salme ships prior to that don't count, nor the Northern Crusades, or pillaging of Sigtuna that occurred later. If it seems a bit arbitrary it's because it is, to an extent.

There was not a single Scandinavian or Viking culture. There were common elements of course, as there still are, but if anything culture were in fact much more local then, than they would be later as the states of Denmark, Norway and Sweden started to grow.

The Viking Age ended with Scandinavia converting to Christianity, which naturally meant a cultural change, but not as complete or sudden a change as people may think. The erecting of runestones largely ended in the late 11th century (although many are christian monuments), but at the same time another monument was born, Eskilstuna-coffins began in that era and were adorned in the same Urnes style as late runestones. It's worth pointing out that the Urnes style, one of the more iconic 'Viking' styles is also named for its appearance on a stave church built a century after the end of the Viking Age. In other words, the style and culture did not stop or suddenly change but developed with time. The Urnes in particular would later fuse with Romanesque styles from the continent.

In the meantime, churches and monasteries were built, learned people started writing in Latin, and the first preserved written documents form Scandinavia start turning up in the 12th and 13th centuries. This due to Swedes, Danes and Norwegians beginning to travel abroad not to plunder nor trade, but to study at European universities, mainly the one in Paris in the first centuries after the Viking Age, later to Prague and Rostock. The countries also were set on a path of consolidating their countries' political power into centrally-organized states in the meantime. Thralldom is out, tithing is in.

In 13th century Iceland, Snorre writes down most of what's 'known' about Norse Mythology. (scare quotes since they get passed off uncritically as fact in popular culture, while there's lots of reasons do doubt how accurate and representative it is. Far far less is known for certain) The first histories were written; the Roskilde chronicles in Denmark and Eric Chronicles in Sweden. Denmark goes off and conquers Estonia, Sweden expands east through Finland (also colonizing Estonia's islands) until they hit Novgorod's border and continue fighting with successor states over that border until 1809. The Teutonic order grabs Estonia from Denmark in 1346, Denmark ransoms the Hanseatic city of Visby in Sweden in 1361 for three big barrels of gold.

Now when it comes to the Hanseatic League (or just Hansa), It was an alliance of cities, not a culture. It was not Scandinavian but a north German organization as well. Its cultural impact was significant, but mainly took the form of spreading Low German language and cultural influence. Most Scandinavian loanwords prior to the reformation are from Low German. But Scandinavia only had two actual Hansa member towns, the aformentioned Visby and Stockholm. And out of those two Stockholm's commitment was rather shallow, as they only attended the Hanseatic Days once (in 1366) and was not governed by Lübeck law, even if the German merchants were well-accommodated. Denmark and Norway had no Hanseatic cities proper, but they had offices (literally, kontor ) in Malmø/Falsterbo and Bergen. Most Scandinavians had no direct contact with the Hansa.

Because of what they did to Visby and shared trading interests, Sweden and Norway allied with the Hansa against Denmark and get the Treaty of Stralsund 1370. This is the apex of Hanseatic power. It's all downhill from here. Sweden Denmark and Norway unite in the Kalmar Union under Queen Margaret, but after her death it is weakened after the Engelbrekt Rebellion in Sweden. Sweden would exit the union entirely a century later after much strife (and the Stockholm Bloodbath) that was largely driven by the fact that Denmark had ambitions on controlling the trade through Øresund while Sweden wanted good trade relations with the Hanseatic League. (Nationalism played its part too, but not as much as later Swedish history would have it)

Earlier in the middle ages, Scandinavian countries had lacked the power to take on the Hanseatic league, should they want to do so (which mainly Denmark did). But their power waned and Scandinavians got their act together. Sweden gained independence from the Union with Hanseatic support, but the new Swedish king Vasa consolidated power and later turns on them, Denmark ends up with a short civil war, but by the mid-late 15th century the Hanseatic League is over, Sweden (with Finland) and Denmark-Norway emerge as powerful states dominating the Baltic Trade. The middle ages are over, the reformation is in, monasteries are out. The Bible is translated, books are printed, and a few people in Gotland and Dalarna are still writing stuff in runes.

In short, that's what went on between 1100 and 1500.