In defence of care homes, yes, they are short staffed, but falls happen anyway. My mum was in the corridor, talking to a carer. Another resident walked up, stuck her hand out and pushed mum.... who went down on her bottom and broke her hip. The carer was right there but it wasn't possible to prevent the incident or stop mum falling. And old ladies break very easily.
I'm not saying it wasn't the care home's fault that your mum fell, but there are never enough staff to stand over every resident at every moment. People can fall just getting up from a chair or their legs suddenly go whilst walking. Unfortunately, dementia does affect mobility. Mum recovered from her broken hip and thereafter walked with a frame. She was reasonably stable but things became more difficult when she 'forgot' the process of moving from chair to bed or from walking to sitting - she couldn't remember how to sit herself on the toilet for example.
As your mum is clearly a 'falls risk' now it would be worth speaking to the home about what measures they are putting in place to (try to) prevent further accidents - they should have a proper plan for this. If they haven't, look for a new place.