This isn't specific to Captain Cook. But when talking about finding islands in the Pacific, most have a large population of seabirds. These birds can be found 2-300 miles away from their home island, well beyond what you can survey from the crows nest of a ship. This in turn gives even the smallest islands a much larger discovery radius. If a boat at sea, finds some birds they can just follow them to land.
As for a source I have seen this fact several times over the years, but I know its in Jack Diamond's Collapse, in the section about Easter Island.
I suppose you could say it was dumb luck. Cook first ran into Kauai as he was on his way north from Tahiti in search of the fabled and long sought northwest passage. After a rather unfruitful journey, he turned back south and decided to check out the islands that he had seen- his stopover in Kauai had been brief. Cook was an accomplished navigator, so finding his way back was little trouble. The second time he found himself anchored off the coast of the Big Island (aka Hawaii); no matter your interpretation of the events that follows; whether Cook was mistaken for a God, interpreted as a God, allowed to pretend to be a God, assumed he was a God, taken as the avatar of a God (though no matter your feeling keep in mind a God in Hawaiian, akua, is not a simple parallel to the Judeo-Christian concept of God); Cook is killed after trying to take the paramount chief/Ali‘i prisoner when he returns to Hawaii shortly after his ships suffer damage in a storm.
As /u/leftovernoodles shared; sea birds will roam extremely far beyond the islands on which they live; most birds fly away from islands in the morning to go hunt and eat and then fly towards the islands as the sun goes down, knowing the flight tendencies of birds can provide a good general direction towards land. Islands; especially the high mountainous islands of Hawaii also create long cloud shadows that can be seen from very far away. Though some of these techniques are usually ascribed to Polynesian navigators rather than Cook; he had intimate knowledge of Polynesia navigation and by his ill-fated third voyage, he had a lot of experience in looking for land-sign and land-finding. So no doubt his crew and him had seen the signs of land well before arriving in Kauai.
Cook's journal seems to indicate this is basically how they came upon Kauai as they traveled north from Tahiti.
On the 2nd of January, 1778, the ships resumed their course to the northward, and though several evidences occurred of the vicinity of land, none was discovered till the 18th, when an island made its appearance, bearing north-east by east.
Cook did have dealings with Polynesian navigators; perhaps the most well documented and well known being the Tahiti ari'i Tupaia. Tupaia gave Cook navigational directions and shared with him the location of a number of islands that he knew about. Tupaia was a very intelligent man- he was a well trained voyager with knowledge of many islands in the region, he was able to share information with Cook thanks to his immersion and position in the body of Tahitian oral knowledge on voyaging and navigation. Even more incredible, Tupaia was able to translate that knowledge into a European style map, drawing the positions of the islands he knew about in relation to Tahiti. The map is not perfect- translating from oral knowledge to cartography is not easy- especially when european abstract representations of geography would have been a novel concept. While virtually all of the islands Tupaia plotted were real and basically where he imagined, even though he had been to only a fraction of them. None of those islands were Hawaiian Islands though.
Some extra reading if you are curious about the Cook in Hawaii moment, Cook in general, or European navigators in the Pacific:
Beaglehole, J. C. The Exploration of the Pacific. Stanford, California: Stanford Univeristy Press, 1966.
Obeyesekere, Gananath. The Apotheosis of Captain Cook; European Mythmaking in the Pacific. Princeton: Princeton University Press, 1997.
Sahlins, Marshall. Islands of History. Chicago: University of Chicago Press, 1985.
Salmond, Ann. Aphroditeʻs Island: The European Discovery of Tahiti. Berkeley: University of California Press, 2010.
Smith, Bernard. European Vision and the South Pacific. Melbourne: Oxford University Press, 1989.
Beaglehole was the Cook expert par-excellence; so his work is definitely a good way to read about Cook (though he has been criticized for sometimes being a bit along the lines of hero-worship). Sahlins and Obeyesekere represent one of the most bitter debates in anthropology with both taking very different views in what happened when Cook arrived in Hawaii.
*edited again to add that you can read Cook's journal for the Third Voyage online to get some idea what he was thinking/seeing when he found the islands. [Check it out.] (http://www.wdl.org/en/item/23/). Though perhaps keep in mind that his journals were heavily edited for publication, especially after his death.