Hello:)
Could you help me?)
The sentences:
1) Buskers can’t play anywhere they want to.
2) What kind of music do you listen to?

Why do we need to use to at the end? Can I just say buskers can’t play anywhere they want or what kind of music do you listen?

Question

Hello:)
Could you help me?)
The sentences: 
1) Buskers can’t play anywhere they want to.
2) What kind of music do you listen to?

Why do we need to use to at the end? Can I just say buskers can’t play anywhere they want or what kind of music do you listen?

Answer

1).   Yes, you can say:   'Anywhere they want'.  I believe 'to' is implied.

2) No, you can't say 'What kind of music do you listen.'

One way to think about this is:   LISTEN just means be aware of, pay attention to, sound.      So, you can say "Listen" if you are referring to this generally activity.   
"Please listen when I am talking."

But if the activity is focused on something specific, i.e. if there is a direct object, there has to a preposition, usually TO.
"Please listen to me when I'm talking."   NOT "listen me"

"Listen to the music. "   NOT  "Listen the music."

Не упустите возможность выучить язык, не выходя из дома. Ознакомьтесь с подборкой опытных преподавателей и запишитесь на свой первый урок прямо сейчас!