Gregor says he would rather starve to death than show his sister that he is hungry, suppressing his impulse to beg her for food in order to avoid inconveniencing her. He also has to watch his movements carefully. If he drops his head, this bothers his family, and so he tries to put his head down carefully. He tries to do everything possible not to bother them. Even though he has been freed from his economic imprisonment, he has become a prisoner in a different way. Literally, he has been locked in, and he off-handedly mentions "his imprisonment." But psychologically he also remains a slave, since he must monitor every move out of concern for his family. Gregor's new prison seems even worse than his old one, though this one is unbearable only because of his own need to serve his family. Gregor is crushed by his inability to help them; hearing them mention money, he "threw himself down on the cool leather sofa, he felt so hot with shame and grief."
When Gregor climbs under the couch, he does so in order to spare his family the pain of having to see him. His hiding, however, is an escapist move. It was escapism that seems to have brought on his metamorphosis, allowing him to find freedom from work. Now, enslaved by his feelings of duty to his family, he attempts to escape these by hiding under the couch and out of the way.
The couch also plays a role in demonstrating Gregor's isolation. Gregor's link with humanity is shattered when no one thinks of attempting to communicate with him directly. The family does not speak to him, and he learns of what is happening only by voyeuristically listening in on their conversations. The couch intensifies this voyeuristic aspect when Gregor surreptitiously watches his sister from behind a sheet while hiding under it.