A wide variety of different drugs can induce musical hallucinations, indicating that there must be some simple non-specific mechanism underlying this. So with bacteria, there will be a simple common process.
Examples of mechanisms, leading to resistance to more than one, perhaps unrelated, class of antibiotics are increased activity of efflux pumps (Buffet-Bataillon et al 2016: quarterny ammonium compounds and fluoroquinolones) or changes in the composition of the cell wall or lipid layer composition (Meena et al 2016: antimicrobial peptide and Alamethicin; Fleitas & Franco 2016: antimicrobial peptides and peptides of the innate immune systems). Another mechanism is down regulation of porins (e.g. Enterobacteriaceae or nonfermentes under imipenem treatment) affecting other beta lactams as well. Combinations of mechanisms (down regulation of porins, increased efflux, development of a mucous cell wall) may end up with multiple drug resistant bacteria.
In addition to what Oliver Nolte mentions, there is the simple explanation that many drug resistance plasmids carry multiple loci each conferring resistance to different antibiotics. So in many cases it is as simple as genetic linkage on a single plasmid.
In addition to Oliver Nolte and Mickael J. Benedik
Refer Obolski et al., 2016: Antibiotic cross-resistance in the lab and resistance co-occurrence in the clinic: Discrepancies and implications in E. coli
This manuscript explained all possible mechanism behind antibiotic cross-resistance