Why did the population of Anchorage, Alaska, increase by so much in the 1970s?

As others have noted, the North Slope oil boom was a major motivator. As somebody who moved to Anchorage in 1976, I'll add a couple more based on personnel observations:

• The raising of oil prices by OPEC was previously mentioned but in the context of increased demand for North Slope oil. Those rising energy prices also contributed to a recession in the mid-70s that made it harder for many to find employment. This was especially true for recent college graduates. However unlike the lower 48, Anchorage had a big shortage when it came to white collar jobs. Recent graduates with a wide range of degrees were snapped up and placed in positions with far more responsibility than they would have had in the lower 48.
• It was still a period when a counter-culture back-to-nature mind-set was very prominent. Alaska was, in this regard, a big draw for many. Simultaneously, many returning Vietnam vets wanted to be "far from the madding crowds". Many of the recent arrivals I met fell into one, or both, of these categories.

Finally, in response to the fellow from Tulsa who mentioned the popularity of the song "Freeze a Yankee", I'll add that a popular joke in Alaska at that time was:

Q: How to you define happiness? A: Seeing a Texan headed south with an Okie under each arm.

What happened was the 1970's Oil Boom (known in non-oil-producing areas as The Energy Crisis).

For various reasons (some political, some economic) in the 1970's world oil production started failing to keep up with demand. This caused major economic hardship in much of the developed world, but was a tremendous boon to areas that were producing oil. They were now selling oil as fast as they could pump it from the ground at much higher prices than historically. There was a tremendous growth in wealth and jobs in any area that supported oil exploration, drilling, or refining.

The period was not uniformly negative for all economies. Petroleum-rich countries in the Middle East benefited from increased prices and the slowing production in other areas of the world. Some other countries, such as Norway, Mexico, and Venezuela, benefited as well. In the United States, Texas and Alaska, as well as some other oil-producing areas, experienced major economic booms due to soaring oil prices even as most of the rest of the nation struggled with the stagnant economy. Many of these economic gains, however, came to a halt as prices stabilized and dropped in the 1980s.

As mentioned above, Alaska was one such area. My hometown of Tulsa was another. We were the fastest-growing large city in the nation for a while during the 1970's. A good indication of the glee (hubris?) this produced in oil-producing areas is the 1978 song Freeze a Yankee, which was popular here on the radio at the time.

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

Population influx

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

Also:

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 Healthcare

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.