Is it illegal to host a website in another country to stream music and why?

Almost every country in the world is a signatory to the Berne Convention which regulates intellectual property, including copyright. This means there is virtually nowhere on the entire planet that will allow you to stream music without prior permission from the owner or that will protect you when you get caught and sued. Theft is theft, even on the Internet.

Illegal as in a crime? That varies from jurisdiction to jurisdiction. Think of the FBI warnings at the start of DVD movies. Copying DVDs is a crime. Copyright infringement is typically a civil violation of the local copyright law of a given country. So - unless you obtain permission - you should NOT do this - as the owners of the music copyrights will sue you for damages. One Boston University grad student was sued for copyright infringement in the US Federal Court in Boston. His penalty was $675,000. Do not do this...