I think as well as the excellent points mentioned above, Romanticism allowed space for a consideration of the supernatural - for example in the poetry of Coleridge. Rationalism, with its emphasis on a solution for everything and man's intellect, explained too much, it could be said, and Romanticism gave room for what we do not understand and celebrated the limits of man's intellect in this sense. Remember Coleridge's purpose in writing the Lyrical Ballads was to make the supernatural seem natural, and this is something that we see in poems such as "The Rime of the Ancient Mariner".
The romantics felt that rationalism was taking man away from nature, and they exalted nature and the emotions, themes, and images that are associated with it. They also thought that while rationalism was supposedly based in truth, it ignored or buried more "essential truths" which romantics valued. For example, a romantic might have said that a myth is more true then historical fact, because myths provide insight into what people thought and felt, and were not just recordings of what some said happened.
Romanticism was a movement which was partially brought about as a response to and rejection of the Enlightenment period. Enlightenment was an intellectual period which argued that reason was the basis of all authority. The Romantic movement stressed strong in emotion rather than rationalism. The Enlightenment Period had seemingly wiped out emotion. Romanticism's attempt was to wipe out logic in favor of emotions. This period produces wonderful poetry, music, and literature, but kind of failed in the social and political worlds because they thought so little of logic and reasoning.
Writers rejected rationalism for the same reason that rationalism was rejected by the movement as a whole- it was in rejection of Enlightenment, which had sucked emotion from writing, politics, art, etc. Writers in the Romantic period favored depicting emotions such as trepidation, horror, and wild untamed nature. Some of the greatest writers of all time came from the Romantic period and presented themes that were anything but rational, rather they were fantastical, wonderfully, beautifully written.