If people originated from Africa, how did they end up on islands such as Hawaii?

TL;DR- people spread across the Pacific over the course of thousands of years as ocean technologies improved; evidence for early settlement dates come from studying environmental changes and archeological sites.

Time and maritime revolutions. This is actually the type of question I think concepts like David Christian and "Big History" are uniquely suited to answer because we can look at the long duree of human experience and migration.

In order to really answer the question, I think it best if we work backwards rather than forwards and start from Hawai'i and think conceptually back to the initial human migrations into the Pacific. I don't know enough about 'Out of Africa' approaches, or the various human migrations that must have occurred before anyone first set out into the great Ocean, but we can work back to at least the beginning of the history of human voyaging coastal seas off Asia.

So first off; Hawai'i has not been inhabited that long; the first settlers arrived perhaps as late as 1000 C.E. or as early as 500 C.E. Before that the islands were home to several species of large Avian megafauna and dominated by large amounts of coastal dry land forest. Besides the giant birds, seals, and fish; there really wasn't a whole lot to eat on the island. We may think of Hawaii as a paradise today, but in the more distant past it would have been a pretty inhospitable place. We owe a lot to the original Polynesian explorers and then settlers that came to the islands for our vision of fruiting trees, taro patches and irrigated terraces- the major foods and crops in Hawai'i were all brought to Hawai'i by these original settlers- who would have had to do a great deal of land clearance and planting in order to create the fertile islands we think of today. So we can date the arrival of the first Polynesians pretty well to between 500-1000 CE; it is during this time period we find the first large scale burning and forest clearance and eventually evidence of crops and cooked animals. We find settlement expanding as we study the changing ecosystems of the islands; humans and their domestic animals (pigs, dogs, chickens, +rat- not domesticated but an important arrival with humans) radically changed the composition of the Hawaiian Islands (and much later on European arrivals and their favored biota would again radically change the environment- perhaps even more so).

These Polynesians that arrived and settled Hawai'i had to come from somewhere; for that we can turn to two possible points of departure- and perhaps both of them were true and the source of different migrations. One source for the Polynesians that arrived in Hawaii is the Marquesas Islands which were settled around 300 C.E.- again this date comes from studying changes in the environment and the early evidence of human habitation- namely the arrival of pottery shards. By about 500 C.E. it appears that the other local island groups; the Society Islands (Tahiti), Tuamotus, Australs, etc. had also been settled. With oral traditions providing some evidence that Society Islanders also migrated up to Hawaii at some point. These Polynesians carried much of the same crops and animals to Tahiti and the Marquesas as they would to Hawaii. Again, there really wasn't a whole lot of natural resources on these islands for people to eat (besides of course marine ones). Polynesians tamed and settled the land, primarily planting taro and breadfruit in order to feed the growing populations and raising livestock usually comprising pigs, dogs, and chickens.

These islanders also had to come from somewhere else; to that we turn to the Polynesian heartland of Fiji-Tonga-Samoa; all of which appear to have first been settled no later than 1000 B.C.E. by an Austronesian people sometimes referred to as the Lapita people because of their distinctive pottery style; a style that seems to have ended around 300 C.E.- partly explaining the date for the settlement of the Marquesas, which must have been settled by the time the pottery ceased to be manufactured. There was at least some amount of long distant trade between these islands in Remote Oceania and the Lapita heartland in Near Oceania (also known as Melanesia) as evidenced by some of their material remains, including pottery and obsidian that can be traced back to Near Oceania. It was on these islands, and the constant voyaging that we know went on between them, that the Polynesians developed long-distance voyaging strategies and technologies; the developments necessary for the later settlement of the Marquesas and Society Islands and then even later Hawaii, Rapa Nui, and Aotearoa (New Zealand). It is amongst these islands that archeologists nowadays argue the culture of the Austronesians settlers developed into a new Polynesian culture- of course these are terms we use to describe the people- not terms they would have used or recognized- nor would they likely have pinpointed a split at which point Austronesian becomes Polynesian- thats a device and distinction completely of our own artificial creation.

These Austronesian/Lapita peoples themselves originated in Near Oceania amongst the island groups sometimes referred to as Melanesia and Micronesia. Here the story gets more complicated; peoples in this area mixed and moved and migrated a great deal. They voyaged and traded with their neighbors and at least sometimes over long distances. We can say that by 1500 BCE then Austronesians had reached the western borders of Melanesia carrying with them their crops and the Lapita pottery style and motifs. The Austronesians mixed and met a number of different peoples in this region from about 4000-3000 BCE until they set off towards Remote Oceania by 1000 B.C.E. There were earlier groups of people who had migrated into Near Oceania.

The Austronesians had likely come from Taiwan around 4000-3000 BCE, taking with them part of their unique agricultural system, and then further adapting and changing in Island South East Asia and then in Melanesia, picking up different crops and animal breeds as they went along and settled and intermarried and exchanged.

The Melanesians who inhabited Near Oceania, and who had migrated into the area previously, likely began settling around 30,000-40,000 years ago, at a time when the islands would have actually been much larger and the oceans shallower. At this point the history becomes murky and unclear. It is unclear when people first started making round trip voyages out to the islands and for how long they made one way trips; nor is it really known what motivation they would have had, but certainly they must have been a maritime centered society. Archeologists generally believe settlement happened in waves; that there was not just one great migration but a series of people coming into the area and gradually spreading out. It must certainly have been a deliberate enterprise most of the time; otherwise why bring plants, animals, and women?

How did people get to South East Asia/East Asia? Not sure; to answer that you will need someone else with a different expertise.

Here is some reading on the subject of the settlement of Oceania, Austronesians, and Polynesians if you are interested;

Irwin, Geoffrey. The Prehistorical Exploration and Colonisation of the Pacific. Cambridge: Cambridge University Press, 1994.
Bedford, Stuart, Christopher Sand and Sean P. Connaughton. Oceanic Explorations: Lapita and Western Pacific Settlement. Canberra: Australia National University Press, 2007.
Bellwood, Peter, James Fox and Darrell Tyron, eds. The Austronesions. Canberra: Australian National Univeristy, 1995.
Kirch, Patrick Vinton. A Shark Going Inland is My Chief: The Island Civilization of Ancient Hawaiʻi. Berkeley: University of California Press, 2012.
Terrell, John. Prehistory in the Pacific Islands: A study of Variation in Language, Customs, and Human Biology. Cambridge: Cambridge University Press, 1986.

edited for spelling