What does brashly smug mean?

To be smug is to be full of yourself, conceited, over confident, etc. It's not exactly a good thing.

And to be brash is to be engergetic, tactless, etc.

So, to be brashly smug would to be very embraceful and outgoing with your over confidence.

An example would be the high school jock gloating nonstop about how great he is at football.