You are correct. I think it's a glitch with how the audio for sunt got cut.
The Romanian phrase is missing an indefinite article as in the English translation. It should read, Eu sunt un bărbat.
Eu sunt bărbat means that I have all the general qualities of a man (biologically, psychologically or both).
Eu sunt un bărbat means that ”I am a man” and it can have more interpretations: I am a man like many others, I am a man (from a specific group of men) etc. In some contexts it might be understood like the first sentence.
So, from en->ro: ”I am a man” can mean both ”Eu sunt bărbat/Eu sunt un bărbat.”
In Romanian, there are general statements that gives a general qualification to the subject like: Eu sunt elev. - I am a pupil. Ea este profesoară. - She is a teacher/professor.
It doesn't miss anything. It is a correct sentence. Don't get confused by similarities of Romanian and English. The Romanian syntax is really different and is much more complicated than it seems (sorry, but it's true).