How does resistance to antibiotics occur?

By mutation: Just like evolution, survival of bacteria that are immune to antibiotics. The genetic structure of the bacteria can change (genetic adaptation) and henceforth could be resistant to a particular or multiple antibiotics. Hence use of prophylactic antibiotics is often discouraged and especially in farm raised animals such as cattle or poultry.

Adaptation: Bacteria, like all organisms, are geneticly adaptive. Repeated or over zealous use of antibiotics can trigger genetic modifications that protect them, rendering them resistant!