Well the twins one can get a trifle complex - there's a simpler demonstration.
The point is that time doesn't work the way we all tend to think that it does. All is happy in the garden in our everyday lives but if you start travelling close to the speed of light time slows down relative to the 'stationary' world.
My favorite example is muons. These little exotic particles are formed in the upper atmosphere as cosmic rays crash into the air.
Now these muons are short lived and only a very few should live long enough to make it down to the ground. But we don't see a few, we see loads - in fact nearly 100,000 times as many as we ought to see.
The reason is that they travel so fast time slows down for them long enough for them to make it to the ground.
http://hyperphysics.phy-astr.gsu.edu/hbase/rel ativ/muon.html
The point of all this is that we may think that we all know how time works but in extreme situations it's actually pretty different.
Bit like a blind man who's only ever felt the trunk of an Elephant and is being told that there's way more to it in reality :c)