Antibiotic resistance comes from selection pressure: any bacteria that are resistant will survive and replicate, so in the next generation of bacteria you will find more resistance.
Bacteria do not simply mutate to form beneficial alleles, the alleles or genes that confer drug resistance must already be in the population; by using antibiotics, however, you're killing off their competition by killing the bacteria that don't have the resistant allele(s)/gene(s).
It's very unlikely that one individual using one antibiotic for one illness will lead to detectable antibiotic resistance. However, antibiotics aren't used in just one individual: they are used repeatedly, in many individuals, often in multiple species, etc. It is this broad use of antibiotics over time that eventually "finds" strains that are antibiotic resistant. Once these strains with resistant alleles become more prevalent, it becomes more likely that they will be present in some quantity to be selected for in the future.