How have bacteria become resistant to antibiotics?

Microbes resistant to antibiotics, disinfectants and other chemicals by a process called natural selection. When we treat a population of microbes with chemicals that we use to kill them, then not all the microbes are the same. They are different in much the same way that humans are different in a crowd; some people are bigger, and may be stronger. Some are fatter and may be able to live longer with out food. Some are thinner and may be able to run faster all get through narrower openings.
In much the same way, some microbes can live off foods that have a microbes cannot digest; some can stand higher temperatures and some can stand lower temperatures. In particular, most populations of microbes are very numerous, many millions all many billions. When you have so many microbes, you can be pretty sure that for any chemical you apply some of them will prove tougher than others. Some might be able to break down the chemical. Some will not absorb the chemical. In much the same way, if you gave a lot of poisonous nuts in very hard shells to a lot of squirrels, those squirrels who could not bite through the shells would not get poisoned. They might starve if they could not find anything else to eat, say soft-shelled nuts or berries, but as soon as those poisonous hard-shelled nuts showed up, the squirrels with the weak teeth would live while the squirrels with the strong teeth would die.
Now, as a rule squirrels with weak teeth would have baby squirrels with weak teeth, so we would find pretty soon that they would not be many squirrels with strong teeth, so we could no longer kill them with hard shelled poisonous nuts. We would have to find some other way of doing it.
That is an example of selection.
It works in much the same way with microbes, bacteria and other germs. If we take a substance like penicillin and we carefully use it only when it is vitally necessary, and we look after it to avoid any spills, then very few bacteria will ever come into contact with penicillin, so most bacteria will be killed quickly if they ever get treated with penicillin. So if someone has blood poisoning from a bacterium like say, Staphylococcus aureus, a quick treatment with penicillin could have him up and healthy in a few days. From then on his body can take over and finish the job. However, if we stop the treatment too quickly, then some of the bacteria will survive, and those will be the ones that withstand penicillin the best. It is sensible to continue the treatment for a few days to make sure that none of the tougher germs live.
Also, it is a good idea not to go using the antibiotic for every little infection, even those infections were the antibiotic will not help. Or for that matter we should avoid using the antibiotic carelessly so that a lot of bacteria come into contact with it, say in sinks and drains, or in the noses and throats of doctors and nurses who keep on breathing in little bits of antibiotics. All those bacteria that keep on getting into contact with little bits of antibiotic pretty soon get selected so that mostly the tough ones will survive.
Another very dangerous practice that should not be permitted is to use antibiotics in animal feed to encourage their growth. Many extremely valuable antibiotics have been made useless by that sort of thing.
So, in general, resistance to antibiotics and in fact to other kinds of pesticides, such as insecticides, is often the result of people using or abusing such tools in ways that favour the pests that are most resistant to them. Such abuses commonly will cause increases in resistance very quickly, sometimes in just a few years. It also becomes possible when bacteria are selected first for resistance to one kind of antibiotic and then resistance to another that we can breed "super bugs" that are resistant to almost all our antibiotics. If we are not careful we might wind up not much better off than before we had any antibiotics at all and millions of people died from infections that sensible use of antibiotics could easily have prevented.