They do not claim to have a "new" way of providing care, but unlike other medical organizations they do not rely on support from governments or institutions so they are able to provide care independently and to those who need it despite the political or socioeconomic factors.
Doctors without Borders has set up their own facilities and supports their own facilities so they do not have to rely on the infrastructure of the country they are in. However, they do heavily rely on private donations so they continue to operate.
This organizations claims not to get involved in the political/ institutions of a country, but to provide medical care and relief to patients impartially. Their approach is to simply provide people with the care they need.
This organization has produced many research articles over the years on the initiatives they have started. Their most recent article is on sexual violence and crisis and conflicts patients face and doctors face treating the victims.
Their staff is comprised of professional and volunteers with a medical background, primarily nurses and doctors in the field but they have many other logistical positions available.
They support their approach based on what they as an organization have already done in countries such as Haiti, Nigeria, Jordan, and Syria.
The aim of this organization is to provide emergency medical care and relief supplies to those who need it. They also work to provide medical care for those trying to escape violence or war.
This organization relys on private donations so that they can operate independently from governemnts or institutions.
Financially they have to secure funding. They rely on private donations that may not always be consistent. They also have to maintain certified and experienced medical staff to provide care.. Finally the security and safety of their staff when they are in areas of conflict or high violence.
While, I can not find any specific events that have motivated their thinking about disaster and health, I believe this organization is motivated by the idea that people should be provided the health care they need despite their social or economic status.