How does a vaccine lead to the production of antibodies against a virus?

The vaccine contains antigens of the virus. When injected, these antigens are recognised by macrophages which engulf the antigen and present the antigens on their surface. Once the antigens are presented, they activate T-helper cells with complementary receptors to the antigen which in turn activate B-cells with specific antibodies to the antigen (clonal selection). These B-cells divide by mitosis to form plasma cells (clones) which produce more antibodies.