"I am the musician."
Translation:Én vagyok a zenész.
"Zenész vagyok" is "I am a musician" - that is what I am is a musician. (Question: what are you?) "Én vagyok a zenész." is "I am the musician" - it isn't him or her it's me (Question: who is the musician?)
The English is an answer to the second question - "Who is the musician?".
That is the verb, otherwise the entire sentence makes no sense. It's the "I am" part of the sentence. What could be dropped is "én", by saying, "A zenész vagyok". This looses the focus from me being the musician that is in the original translation. Depending on context one or the other would be more appropriate.