How is your horse housed? Is she boarded? If so, it is probably too late to avoid spreading it to other horses but extreme care must be taken now that the abscess has opened. She should be fed last and if possible only handled by one person who uses a foot bath and does not handle any other horses after her.
Strangles is extremely contagious, especially once the abscesses open up. As far as how she got it, if there is a history of it on the farm any of the horses could be asymptomatic carriers.
For prevention, there is a vaccine but it isn't extremely effective. It does reduce the risk and vaccinated horses who come down with Strangles often have only a mild case of the disease. There are a lot of risks associated with the vaccine and so it should be a case by case decision whether or not to give it. I would DEFINITELY call your vet back and ask how he/she thinks you should manage your mare to avoid infecting any other horses (since your vet will be familiar with the layout of the barn, etc. they will be able to come up with a good quarantine plan).
Antibiotics are generally not given to treat Strangles unless the horse is extremely ill. Once the abscess closes over she should start feeling better but will be contagious for up to a month and will always have the possibility of becoming a carrier horse.
If the vet seriously considers it to be strangles, I am a bit puzzled as to why he hasn't come to take a look and to quarantine! Strangles can run a muck through a stable. It is generally just a nasty infection that spreads like wildfire...usually leaving your horse off work for quite some time. It rarely kills but if left unchecked, it can become dangerous. Do you board? If so you better tell your BO that your horse may have strangles. They need to be seperated!
My BO doesn't think its strangles. She doesn't have any other symptoms besides the lumps.