A few of the highlights include some really interesting and insightful professors and classes, field studies which take you to places around Copenhagen you would never normally get to see (such as a debate in Parliament, the ballet or the former working class district of the city) and the long study tours, which give you and your classmates the chance to spend a week in an awesome European city or cities to see the sights and at the same time learning about your class' topic (my international law class went to Sarajevo in Bosnia and Herzegovina and had an unforgettable experience meeting local people including members of the EU,NATO, professors, leaders of the local religious communities and survivors of the war, which would have never been able to do otherwise).
DIS is there for you to use as much as you'd like - you can get their help everyday or never once talk to them, up to you.
Housing can vary, I talked to people who had good and bad experiences with host families, in kollegiums and in DIS housing so it all depends. My home stay was excellent, and although it limited me socially (especially for going out on weekend nights) since I lived so far outside of the city, their good cooking, the great dinner conversations, insight into the culture and having a family and a house to come home to every night was the highlight of my time in Denmark.
Some negatives: Denmark is one of the most expensive countries in Europe. I'm talking $8-$10 a beer expensive. You will spend less with a host family since they provide most of your food, if money's a big concern, I'd be hesitant to spend a whole semester here. Also, Copenhagen is an expensive and limited place to fly into and out of, and since DIS's off-day is on Wednesday and you therefore have classes on Friday, the policy is to discourage weekend traveling. So if you were planning to visit a different country every weekend, I'd look at other bigger travel hubs (like Milan). Also, DIS is comprised of 800 students, and I've heard they are planning to accept even more students next semester, and I definitely felt overwhelmed by the number of fellow American students. It takes away from a more personal experience meeting people and in classes, and it also made it very easy to get comfortable with American students and friends rather than venturing out on your own, meeting Danes, immersing yourself in the culture and becoming more independent and self-reliant.
Overall, the home stay, the long study tour to Bosnia and Herzegovina, the profs, field studies and classes all outweighed the negatives and made this a great semester for me and showed that DIS is a really great program if you navigate it the right way.