Also be aware that in some contexts this can have sexual connotations.

I don't know why we say 'big head', but it is a common idiom. And it does not mean some REALLY has a big head (in size). It just means someone is very arrogant. You might say, maybe in a work environment, 'His head is so big he won't fit through the door', to describe a very arrogant colleague. We have another expression 'big wig'. This describes important people within a company, such as top management. I believe this expression comes from very long ago when the elite of Europe wore tall wigs. Hope this helps. :)

Someone who is arrogant thinks they are better, smarter, etc. than other people. All of this is inside their head, and they think this more than other people, so their head must be bigger to fit their ego. Thus, they have a big head. I hope this helps. :)