An oil-filled space-heater radiator is a freestanding version of what was under each of Grandma's windows, connected to a boiler for hot water or steam. Like Grandma's radiators, it's actually also mislabelled.
Convection, Not Radiation
Both freestanding and system "radiators" actually heat by convection --- the heat from inside the coils or fins is transmitted to the outside surface by conduction, and as cool air contacts that hot outside surface, it rises into the room and draws more cool air into contact. In practice, this means these heaters don't need fans to distribute the air they've warmed.
Heating What's Inside
The oil inside a freestanding radiator doesn't burn. It's used because it doesn't need to circulate through a boiler to be heated, so it can be heavier, and because it's thicker than water, it holds heat longer. Electrical circuits warm the oil by actual radiant heat.
Slow Start, Long Finish
It takes some time for the oil to get warm enough to get the convection going, so an oil-filled unit isn't the way to take the chill off a room fast. On the other hand, once the oil is warm, convection will continue for some time even after you've turned off the power.
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