How are antibiotic resistant bacterial infections treated?
There are a number of ways to address antibiotic resistance in infectious bacteria. What one gets used depends on exactly what organism we're talking about. But below is a short list of some ways:


Changing the antibiotic being used. Not all antibiotics have the same molecular target within a bacterial cell. Some interfere with the formation of the cell wall, others interfere with protein synthesis, some mess with DNA replication, etc. Similarly, resistance mechanisms don't work equally well on all antibiotics. So often the solution is simply to switch antibiotics. Or, quite often, treating all infections with a combination of drugs that should get around most resistance. This is why you see people getting skittish about things like multi-drug resistant organisms - because it removes this solution.
This isn't as much a treatment as a preventative measure, but increasing antimicrobial stewardship and the appropriate use of antibiotics to prevent widespread resistance from arising in the first place.
The use of non-antibiotic antimicrobials. The most "common" of these is phage therapy - using preparations of bacteriophages that naturally prey on the infecting pathogen to help combat the infection. This was a major "contender" for how we treated antibiotic infections, but has fallen away due to a bad reputation early in its history (due to difficulties purifying phage treatments) and the discovery of antibiotics which were vastly easier to use. But there's still ongoing research, and it may pose a potential to help treat highly resistant infections in the future.