Why is the air inside an igloo warmer than its outside?

An igloo is not made from ice, but from compressed snow. Snow is basically semi-frozen water or frozen crystalline water.

Contrary to common intuition, snow has very good insulating properties. Solid ice on the other hand, is not a good insulator compared to compressed snow. This is because ice is actually solid but snow is filled with minute pockets of air. While snow on an igloo does indeed look solid, up to 95% of it is actually air trapped inside minute crystals. Because this air cannot circulate much inside these ice crystals, heat becomes trapped inside it.

Some engineering also goes into the design of the inside of an igloo. The inside is divided into levels, where the upper level is for sleeping, the middle one is for fire and cooking (yes, fire! and a little hole is built into the top of the igloo to prevent smoke inhalation) and a lower level is used as a sink for cold air. As we know, heavier colder air naturally drops, and since the lowest level is where the door is placed, this cold air stays there. And warm air which rises, collects where it is mostly needed - in the eating and sleeping levels. And since the entrance to the igloo is at the bottom part - the tunnel to crawl through whilst entering or exiting the igloo - freezing air cannot blow directly into its interior.

Temperatures can reach as low as -50°F (-45°C) outside the igloo but the temperature inside can be a “comfortable” 20°-70°F (-7° to 20°C) (when you’re exposed to those low temperatures outside, coming back in can be most pleasant!).

All of this can be explained if we consider heat transfer and convection. This is a process whereby when a fluid moves, it transfers heat along with it. When this fluid is stationary, it will transfer heat by thermal conduction which is the transfer of heat from one body to another when they are in contact. You can see an example of this when you touch an ice cube and seeing it melt right where you fingertip is. But the more a fluid moves, the greater is it’s Reynolds number since the flow patterns become more unpredictable. The greater the Reynolds number, the more heat that is transferred via convection. And because snows has a low thermal conductivity, as mentioned above for air and how the snow contains air pockets, an igloo stops the heat transfer into its outside surroundings. The compressed snow and stationary air both act as surprisingly effective insulators.

Although igloos are often associated with all Inuit and Eskimo peoples, they were traditionally used only by the people of Canada's Central Arctic and Greenland's Thule area. Other Inuit tended to use snow to insulate their houses, which were constructed from whalebone and hides. Snow is used because the air pockets trapped in it make it an insulator. On the outside, temperatures may be as low as −45 °C (−49 °F), but on the inside, the temperature may range from −7 to 16 °C (19 to 61 °F) when warmed by body heat alone.

italics mine .

Certainly they would use fire to heat also.

The crucial concept is that snow is an insulator.

Because it is heated up from the inside by people (or possibly small fires).

The snow walls work as a thermal insulator to the outside.