They are Vibram Five Fingers KSO. The idea behind them is that they mimic walking and running barefoot as much as possible whilst protecting your feet from grime and broken glass etc.
Yes, they look very odd.
Yes, they are actually quite tricky to put on.
And yes, they are actually very comfortable to wear.
I love to be barefoot as much as possible, and current thinking is that it's better for your foot to use all those bones and muscles as they were designed! Certainly a lot of the people that I have trained with swear by them ....