This is the second definition of smart from OALD:
2 (especially British English) (of clothes, etc.) clean, neat and looking new and attractive
-They were wearing their smartest clothes.
What does neat mean in the definition when used to describe clothes? In the same dictionary, the most relevant definition of neat seems to be def. #1:
1 tidy and in order; carefully done or arranged
However, it feels like to me that this sense of neat is more appropriate when referring to things like your desk, office, room, rows of books, etc. in which it is arranged properly and in order; where as if the term is used with clothes, it doesn't seem to make sense because how do you arrange an article of clothing and make them in order? I mean you could iron it to make it smooth, but you don't arrange it?