This is super easy. In every population you have differences between the individuals in that population. If you apply a selection pressure to that population (like antibiotics to bacteria), you will kill all the bacteria that do not have the ability to withstand the drug. Thus, you have selected for those bacteria that happen to have a better ability to withstand it.
Antibiotic resistance is a consequence of evolution via natural selection. The antibiotic action is an environmental pressure; those bacteria which have a mutation allowing them to survive will live on to reproduce. They will then pass this trait to their offspring, which will be a fully resistant generation.
Survival of the Fittest (Natural Selection):
When bacteria are initially exposed to an antibiotic, those most susceptible to the antibiotic will die quickly, leaving any surviving bacteria to pass on their resistant features to succeeding generations.
