Before the 19th century, medics serving in wars are recognized depending on the country they are serving and that makes for a lot of variations. A Swiss citizen named Henry Dunant was traveling to Solferino when he saw thousands of soldiers left wounded and uncared for during the War of Italian Unification. He wrote a book(A Memory of Solferino) proposing to set up in peacetime and in every country volunteer groups to take care of casualties in wartime and to get countries to agree to protect first aid volunteers and the wounded on the battlefield.

With this proposal, a committee (which would become the International Committee of the Red Cross) was formed to adopt a single distinctive symbol backed by the law to indicate respect for army medical services, volunteers with first aid societies and the victims of armed conflicts. The symbol must be simple, easily identifiable and similar to friend or foe and it must be recognized universally.

Since the emblem was to reflect the neutrality of the armed forces, medical services and the protection conferred on them, the "Red Cross" symbol was used and is basically a reversed version of the Swiss flag.

Switzerland's permanent neutral status has long been established and recognized (Treaty of Vienna and Paris, 1815). The contrasting colours can also be easily recognized at a distance and the emblem can be produced easily. The white flag was and is still a symbol of a wish to negotiate or surrender.

The website of the ICRC has a detailed history about the Red Cross, Red Crescent and Red Crystal emblems.

Follow-up question: How well have this symbol been respected since it surfaced, in the various conflicts it was present?