There is a cultural background to this, but there's also some interesting recent economic factors.

Pacific Islanders have always had an acceptance of weight; the wealthier you were, the bigger you were. But before colonisation, the general population weren't overweight; their diets were largely foods like taro (breadfruit), coconuts and seafood. These are all labour-intensive sources of food - coconut and taro take a lot of physical preparation to get ready (have you ever had to harvest a coconut right from the tree??) - and collecting seafood with no metal or plastic implements / technology is also hard work. So their diets and lifestyles kept them healthy by default. They'd have a feast once a month or so when someone slaughtered a pig, and eat as much as they could then, but generally they were in good shape. (Here's a photo of a bunch of Samoans from 1890 or so, as an example.)

After trade routes were established, though - and particularly after the intensive Western colonisation of the Pacific during WWII - they started getting imported food. Foods like corned beef and sugary white bread that the soldiers and sailors bought with them became local favourites.

Then, a more sinister development; once refrigerated shipping became ubiquitous, farmers in America, Australia, the UK and New Zealand all discovered that Pacific Islanders had very low standards when it comes to meat. They could take all the fatty offcuts and sell them to these remote islands, for slightly more than they would make selling it as pet food. So unhealthy cuts like chicken frames, turkey tails and mutton flaps from New Zealand lambs (comprising around 30% fat by weight) end up as primary meats in the Pacific Islander diet.

Since they didn't have a history of cattle farming there - the only local non-fish meat came from the occasional pig or goat - they were (culturally) unaware they were getting fed the scraps off the global table. Combine this food with a shift from an active lifestyle of fishing + farming to a Western lifestyle of sitting in shops, in front of computers, and in front of TVs - minus all the walking / public transport commuting that is probably the only exercise keeping a bunch of us from being obese - and.... boom. 9 of the 10 fattest nation-states are Pacific islands.

http://www.independent.co.uk/news/world/australasia/pacific-islanders-fatal-diet-blamed-on-kiwi-exports-655190.html

TL:DR; Pacific Island countries have been used as a dumping ground for fatty meat offcuts from the Western world for the last 70+ years; their obesity epidemic is not entirely of their own doing