Why does Gregor cling to the picture of the woman in furs on the wall?
Gregor clings to the picture on the wall because he is clinging to his past. His sister, Grete, comes in and tries to move all his furniture, thinking it would be easier for him to crawl around the floor and walls if it were gone. Gregor likes the furniture because it reminds him of when he was normal, but he can't do anything about the fact that his sister and mother are removing it. In an attempt to do something, he climbs the wall and clings to the picture so his sister cannot remove it. Ironically, he displays obvious bug-like characteristics in order to cling to his human past.
Another aspect of criticism on this story suggests that Gregor clinging to the picture of the woman in furs symbolizes his sexual frustration, the manliness that his family and job stole from him (and that he relinquished) by working in a job he found meaningless while his family, especially his father, took advantage of him. That the woman is wearing furs in the pictures signifies the intimacy with which he clings to her; indeed, some scholars find this intimacy an act of violation.
Gregor doesn't particularly like the picture, but it does represent to him an element of normalcy and his life as it was before his change. It is the one thing his sister and family do not take from the room, and he clings to it as if his very life depended on it. In his mind, if all things from his human life disappear, the likelihood that he will return to normal is less certain.