Mmm-mangosteen! Many call it the Queen of Fruit, including Queen Victoria, and once you taste one you'll realize why.

Why They're Not Your Favorite Fruit, Yet

I think the only reason the mangosteen has yet to take the world by storm is that it's a terribly inefficient fruit: it may be the size of a mandarin orange, but the actual edible portion is rarely larger than a chestnut, and (even here in Singapore) each one of these armor-clad chestnuts can cost you up to a Sing dollar. A ripe mangosteen at room temperature lasts only for a few days, and refrigeration or freezing quickly causes damage; the only way to store one for a longer time is at 10°C, which will keep it in good shape for several weeks. And finally, when unopened they're not terribly attractive: given their hard surface and mottled red-purple-black exterior, you might be excused for thinking you've stumbled on a pile of dirty beetroots instead. But don't let this discourage you!

Obtaining a Mangosteen

They're in season from April to September in South-East Asia, and are occasionally spotted in the "obscure and overpriced tropical fruits" shelf of your local hypermarket elsewhere as well. But they're worth the effort to seek out, just so you'll know what to look for if you ever do end up in its endemic region.

Opening a Mangosteen

If you have a knife, make a shallow cut (less than a centimeter) around the fruit, starting and ending at the stem, and then twist with your hands. One half will pop off and the inner white flesh will be revealed.

If you want to impress people with your l337 mangosteen sk33lz, here's how to open one with your bare hands. Put the fruit on a flat surface, stem upwards. Interlock your fingers and squeeze the fruit with the bases of your palms. The rind will tear and pop open, and now you can twist/peel the rest off easily without damaging the flesh. Not such a pretty method, but very functional.

And finally, be careful when opening: the rind is usually dry, but if those few drops of its beetroot-like juice touch anything, they will permanently stain it purple.

Eating a Mangosteen

The part you want to eat is the inner white flesh, consisting of 4-8 segments that look quite a bit like a bulb of garlic. The red rind is terribly bitter, so be sure no flecks remain. Larger segments may contain seeds, which are inedible unless boiled or roasted (and not worth the effort), so just spit them out.

The taste of the mangosteen is unique and addictive, which also makes it difficult to describe. Perhaps something along the lines of a cross between a mango and a peach: it's like the utterly delicious first bite of a ripe mango, but counteracted with a tart acidity that stops the sweetness from becoming overwhelming. The taste also depends on the ripeness of the fruit, with younger ones more acidic and older ones sweeter.

The Chinese think that mangosteen are a cooling fruit, and thus the perfect antidote to heaty fruits like the mangosteen's hubby durian, the King of Fruits. Due to the labor involved in extracting the edible bits, mangosteen are not particularly amenable to further processing; just serve them as is for breakfast, snack or dessert.

References

Personal experience
http://facethewall.com/fruit-mangosteen.html