If you're moving Great Works from your city to another city, in the "your culture" interface with different great works slots, click on one and then click on an empty slot in another city. If you want to get a great work from another Civilization in exchange for one of your own, go to the "swap great works" interface. Then, select one of your own great works in the appropriate list for your Civ (one for Writings, Art and Artifacts) and then click on one of the desired artifacts from the same column. At the bottom, you should see a foreign Great Work and your own selected. Click the "swap" button, and move the newly acquired great work to the desired slot.
Welcome to the forum, by the way.