University of Alaska, Anchorage published a study into People and Economy of Alaska, which can be viewed here.
From that source, I'd say there are following reasons for their population boom:
Discovery of Oil
As Tom has already mentioned, Oil was discovered in 1968 in Prudhoe Bay oilfields. Alaska collected $55 billion in oil revenues through 2001, with the peak from 1980 to 1985. Alaska's government spent it on services sectors and infrastructure, which created more jobs therefore providing opportunity to people to give birth more (As they could afford babies).
Quoting from the paper:
In 1968 Alaska had an enormous piece of luck: the Prudhoe Bay oil
field, the largest field in North America, was discovered on land the
state government owns. Much of the social and economic change in
Alaska since then can be traced to North Slope oil development
With events like the oil discovery and other economic benefits, people moved to Alaska from elsewhere, increasing its native population. From the paper shared above:
People still move into and out of Alaska with economic booms and
busts. But more of those drawn to Alaska during the booms of the 1970s
and 1980s stayed on when the economy slowed, giving Alaska a more
stable, older population with fewer young adults
Alaskans have paid no personal state taxes since 1980, and for many
Alaskans, Permanent Fund (A saving account worth 75 billion USD set up
by the state) dividend payments from the state now exceed the local
taxes they pay
Better economy meant better healthcare services for the population which decreased the mortality rates, especially among infants, and increased the birth rate.
Quoting from the same source:
The Alaska Native population doubled in 30 years, as improved health
care helped people live longer and reduced infant mortality. But that
growth has implications for rural Alaska, where jobs are scarce,
incomes are lower, and housing and utility systems are costly
Population concentration in Urban areas
Population in Alaska is mostly concentrated on three urban areas, which include Anchorage.
Nearly three in four residents lived in and around Anchorage,
Fairbanks, and Juneau by 2000. Most dramatically, the share of Alaska
Natives in the large urban areas increased from 17 to 32 percent from
1970 to 2000
Pictured below, a timeline of Alaskan population:
For further details, please read the study.