There are multiple phases in Gregor's family's behavior after his metamorphosis. First, they are horrified, except for Grete, his sister. Then, they discuss their financial situation which worsened since Gregor stopped providing for them. Then, Gregor realizes that his father had even had some savings, which he kept a secret. Gregor learns all this by eavesdropping on their conversations. Finally, he finds out that they have all got tired of him. This is what causes his death. More precisely, he decides to die, so as not to cause more trouble for them by his mere existence.
Gregor is confined to his room and completely incapable of any action, while his family members go on with their lives. Soon enough, it turns out that they don't need him anymore, not that he is a useless cockroach. This way, Kafka conveys the idea that Gregor had put his family's interests before his own. Even before his transformation, he worked the job he hated just so he could provide for his family. Once he turned into a vermin, not only he couldn't provide anymore, but he was also a burden for all of them.
