How do viruses reproduce?
First the virus enters the cell, releasing RNA. The virus RNA invades the cell nucleus and takes over. The virus RNA uses the cell to create new RNA and assemble more viral particles. The new viral particles are released, sometimes destroying the cell in the process.
How does this reproduction lead to illness or disease?
Viruses infect different types of animals, but the same virus can have very different reactions with different types of animals. For example the flu virus can infect both birds and humans. While some flu viruses don't harm birds, they can overwhelm and kill humans.