Once inside the cell, the viral enzymes take over those enzymes of the host cell and begin making copies of the viral genetic instructions and new viral proteins using the virus's genetic instructions and the cell's enzyme machinery (see How Cells Work for details on the machinery). The new copies of the viral genetic instructions are packaged inside the new protein coats to make new viruses.
Once the new viruses are made, they leave the host cell in one of two ways:
They break the host cell open (lysis) and destroy the host cell.
They pinch out from the cell membrane and break away (budding) with a piece of the cell membrane surrounding them. This is how enveloped viruses leave the cell. In this way, the host cell is not destroyed.