Christiansborg is currently used as the home of the Danish parliament and in old days there was a wish from both the king and the parliament to have a certain distance between the monarch and the parliament. This was particularly important up till a century ago when the Danish kings did interfere from time to time in the Danish politic. Today the current monarchy is a constitutional monarchy where the queen has no political influence so the need to keep the queen separate from the parliament is less important but you have kept the tradition of the royal family to have their residence away from the parliament.
Despite the separation of the royal family from the parliament the royal family still got some rooms at the castle which are used at special occasions including state banquets and the annual New Year audience where the queen receives New Year greetings from the higher positions of the Danish society.
You can visit the large rooms at the palace on a tour which will take you around the different rooms. There are two tours a day one in Danish at 11 and at 3 pm you can join an English tour. You do not need to book a tour in advance and can just show up for a tour and then get the tour of the grand rooms.