The basic idea behind any toaster is simple. When you put your bread in and see the coils glow red, the coils are producing infrared radiation. The radiation gently dries and chars the surface of the bread.
Nichrome wire is an alloy of nickel and chromium.
The nichrome alloy does not oxidize when heated. Iron wire would rust very quickly at the temperatures seen in a toaster. The nichrome wires would connect directly to a plug. You would then plug in the toaster and watch the bread. When the bread became dark enough, you would unplug the toaster.