It is a known fact that the island microstates of the Pacific ocean where inhabited for centuries before the Europeans landed ashore and greeted the natives. It is obvious that the Austronesians colonized these parts of the globe by just sailing and island hopping, but I am searching for more specific details.
For example, was it a find-land-or-die-trying thing (they set off into the horizon hoping they would find new land)? It seems completely absurd just how they managed to reach even the easter islands (off the coast of Chile), a microscopic SPEC on the Pacific ocean and even lived there. Assuming they did not have the vessels the Europeans did during the age of discovery, and knowing even the Europeans had huge troubles around transoceanic voyages, I can't possibly think just how Austronesians could carry food, water, and enough population to complete such voyages and even colonize an entire island.
You might want to check out the Pacific Islands section of the FAQ; the "Origins" and "Pacific Islander culture and navigation" sections have answers that touch on different parts of this, mostly by /u/b1uepenguin.
It was generally not a "find land or die trying" thing. Though there was somewhat of a movement to try to paint the polynesian voyages that way. In actuality Pacific Islanders had sophisticated navigation techniques and not only understood how to explore, find new islands, and build up local self-sustaining agriculture on those islands but also to routinely travel between islands for trade and so forth.
A lot of the difficulties of navigation are hugely overstated by detractors. For example, if one is voyaging between two sets of islands it's not necessary to make a pin-point land fall at the end of the journey, rather all you need to do is make sure that you don't miss the entire destination island chain to one end or the other. A modern analogy would be Robert Ballard's techniques in finding lost wrecks on the ocean bottom. He wouldn't look for the ship first, he'd look for the much larger (in terms of area) trail of debris (made up of smaller individual objects). Finding any part of that trail would make it possible to trace it back to the shipwreck at one end of it. Similarly, spotting any island in an island chain means that you have now solved the hard problem of navigating between distant island chains and have transitioned to the vastly easier problem of navigating within an island chain, often where islands are short voyages apart from one another (even perhaps where you can always stay in sight of land when traveling from one island to the next). Additionally, they developed techniques for being able to recognize the presence of an island even when the land itself was over the horizon (through patterns in ocean currents, for example).
It's worth noting that provisioning for those European ocean expeditions seemed like a giant endeavor because they had large crews and didn't expect to spend a lot of time on land between individual trips. If you plan well, take the best advantage of the winds, and have knowledge of the local geography then making "long" trips through the South Pacific is totally doable without taking on an excessive burden in provisioning. Captain Cook, for example, took multiple several hundred ton vessels that were over 100 feet long, loaded with cannons, each of which had a crew of about a hundred. Compare that to a typical voyaging canoe that might be 50 or 60 feet long, with a draft of 10 tons at the high end, and a crew of maybe a dozen. That's literally one to two orders of magnitude easier to outfit and supply, more so if you're not trying to keep a huge reserve of supplies (not just food but gunpowder and trade goods) in between voyages.
This has been borne out by various modern sailings using either modern boats or ancient designs (such as Hokule'a, Kon-Tiki, etc.) For example, just last year the Hokule'a (a recreation of a Polynesian double hulled voyaging canoe) made a 1900 nautical mile sailing voyage across the Pacific from the Galapagos to Rapa Nui (Easter Island) in 16 days. Provisioning a crew of about a dozen for just a little over two weeks is not a terribly onerous feat, and essentially all other voyages around the Pacific would have been shorter and easier.
An excellent and highly accessible book to learn more about the history of Pacific Islanders is Vaka Moana (edited by K.R. Howe), it does a particularly good job of covering their technology and voyaging capabilities.