As bacteria are exposed to the same antibiotic over time, they basically develop immunity to it. There was always one or two bacteria in the population containing the gene for antibiotic resistance due to a mutation, and after said antibiotic is continuously exposed to the population, only the one or two bacteria with the resistance gene will survive (because of natural selection). Those few bacteria left will asexually reproduce, making exact copies of themselves, which, as you may have guessed, contain the gene that makes them resistant to the antibiotic. Over time, a new population will build up, all containing the gene, making the bacterial population immune to the antibiotic.