When teens go to high school only with people who look like them, they are not ready to go to college or to work in the real world. Also, if their families are rich and conservative, or poor and uneducated, they need to see that their are good people who have a different background.
Yes, cultural diversity is important in high schools because people need to see people from different backgrounds to get a full feel for what the world is like. If people don't get cultural diversity in school, then they will not understand what it is like to live in a different background.
The only diversity that really seems to matter in practice for making our society better is diversity of thought. Different ideas and perspectives, breeding critical thinking.
But I suppose this depends on what you mean by cultural diversity. Does this mean some bizarre attempt to try and force more people of different cultures into your schools? Does this mean diversifying the curriculum to show people more about world cultures? Does this mean (what I have seen in some schools) picking pieces of the curriculum at random to have 'cultural' content at the expense of quality content?
All in all, exposure to other cultures is good, but so called forced 'diversity' usually ends up coming off as racist or pandering. Ingenuine at any rate.
I would say that cultural diversity can be a good thing, but it's absolutely not necessary for anybody. People stating that they should be exposed to people from different backgrounds so they know what the world is like, are fooling themselves. You do not gain any great understanding of the world based on what a few ethnic kids at your high school are like. That's like saying if you met a black guy when you were young, you'll understand what all black people are like, and it's born out of liberal ignorance.
People go to schools with 100% uniform diversity in India, China, Africa, and many places across the world. Diversity is not something that should be FORCED.
I live in South Africa, and here applicants of certain races are favored in the admission process because they want the universities to represent the national demographics. This is unfair on people of non-designated races. There are the effects of apartheid to be taken into consideration, but the system is continuing even though most of today's applicants were born after apartheid.