Why is it a little odd to say "I work at nights"?

Answer 1

It's idiomatically correct to say either of the following, but not both:

I work at night.

I work nights.

Answer 2

Idioms and idiomatic expressions can't be explained very well.

http://www.answers.com/idiomatic

Here are a couple of websites with lists of English idioms. Remember ... these are correct expressions, but usually cannot be explained.

http://en.wikipedia.org/wiki/List_of_English-language_idioms

http://eflnet.com/idioms/idiomlist/all