Beta-lactam- and in particular carbapenem-resistant Enterobacteriaceae represent a major public health threat. Despite strong variation of resistance across geographical settings, there is limited understanding of the underlying drivers. To assess these drivers, we developed a transmission model of cephalosporin- and carbapenem-resistant Klebsiella pneumoniae. The model is parameterized using antibiotic consumption and demographic data from eleven European countries and fitted to the resistance rates for Klebsiella pneumoniae for these settings. The impact of potential drivers of resistance is then assessed in counterfactual analyses. Based on reported consumption data, the model could simultaneously fit the prevalence of extended-spectrum beta-lactamase-producing and carbapenem-resistant Klebsiella pneumoniae (ESBL and CRK) across eleven European countries over eleven years. The fit could explain the large between-country variability of resistance in terms of consumption patterns and fitted differences in hospital transmission rates. Based on this fit, a counterfactual analysis found that reducing nosocomial transmission and antibiotic consumption in the hospital had the strongest impact on ESBL and CRK prevalence. Antibiotic consumption in the community also affected ESBL prevalence but its relative impact was weaker than inpatient consumption. Finally, we used the model to estimate a moderate fitness cost of CRK and ESBL at the population level. This work highlights the disproportionate role of antibiotic consumption in the hospital and of nosocomial transmission for resistance in gram-negative bacteria at a European level. This indicates that infection control and antibiotic stewardship measures should play a major role in limiting resistance even at the national or regional level.