The sages prohibited putting food on an open flame on Shabbat, even if the food has already been fully cooked. This is for two reasons: (1) A person might increase the level of the flame, thereby violating the prohibition of burning on Shabbat. (2) Since cooking is usually done over an open flame, an observer might suspect that the person is cooking the food.
At the time of the early sages, in order to return a pot to the oven it was necessary to conceal the flame with straw or ashes. Nowadays that gas is used for cooking, it is possible to cover the open flame with a large metal sheet, as long as the control buttons are also covered (or removed) in order to prevent raising or lowering the flame on Shabbat. The most convenient equipment to use is a covered electric heat source, as in a Shabbat hotplate. To be suitable for Shabbat use, such a hotplate should not have a temperature selector switch or a thermostat. The Zomet Institute issues approval certificates for hotplates and ovens with a Shabbat option after testing them.
Is one permitted to put cold (solid) food on a hotplate on Shabbat? There are different opinions about this. Some feel that it might give "the appearance of cooking" and therefore insist that the food must be placed on top of another pot (Shemirat Shabbat K'Hilchata). Others are more lenient and allow putting a simple barrier (such as an overturned jar cover) between the food and the hotplate. And others still are even more lenient, based on the opinion that since a hotplate is usually not used during the week there is no fear of "the appearance of cooking" and it is permitted to put (solid!) food directly on a hotplate (Rabbi Ovadia Yosef).