Because "bărbat" means man, and "bărbatul" means the man, so using "bărbat" it would say "man eats bread" when you need it to say "the man eats bread". In other words, you need to use the article. Sorry this isn't very well explained, but hopefully you get the meaning now.
In normal Romanian usage, if you simply want to state that you are (a) Romanian, drop the "a". The correct usage is:
(Eu) sunt român.
"Eu" in paranthesis, because it is optional.
You would use un român either to specify that you are one Romanian among others of a specific kind, or when it is qualified with an adjective ( Sunt un român mândru / I am a proud Romanian ).
I'm curious what is that clapping sound when they say the word mănâncă?