Islands in Europe, Africa, and the Americas seem to have been settled relatively late in human history. Madagascar, Madeira, Iceland, and Jamaica were all settled only in the past 2000 years, despite being relatively close to continents.
In Asia, Taiwan and the Philippines seem to have been inhabited for tens of thousands of years. Why were they discovered so much earlier than islands elsewhere?
Quite simply, they weren't islands tens of thousands of years ago. That was near the end of the ice age, when Taiwan and the Philippines were connected to continental Asia. It looked more like this at the time. You can see Taiwan there in the upper right. Places like Taiwan weren't islands, and even more remote places like the Philippines were still easily accessible by current standards.
Meanwhile here's a map of the globe around that time. It's a little hard to see there but you can see that not everywhere that is now an island was once part of a larger continent. It has to do with what are called continental shelves which, in many places, were exposed during the ice age. In places where the climate was more favourable, people would be able to migrate to places that are now more geographically isolated.
It's also worth noting that Taiwan's only placed at around 8000 years ago by fairly recent scholarship (i.e. Blust 1999). The Taiwan Straights aren't that deep, and while it's possible that there was water there (we know not the exact dates of either the first major migration nor of the actual coastlines at every given point in time), it was less significant and still easily crossable.
I can't tell you about islands in Africa or Europe, but then assuming it's the case that they weren't settled, I'm not sure asking why they weren't will produce much in terms of a satisfactory answer except that "nothing happened to cause them to be". But someone else will have to address that.
