The fireplace mantel or mantelpiece, also known as a chimneypiece, originated in medieval times as a hood that projected over a fire grate to catch the smoke. The term has evolved to include the decorative framework around the fireplace, and can include elaborate designs extending to the ceiling.
Originally, as you say, it was to support a hood over the fire. Today it serves to cover the joint between the fireplace itself (usually lined with firebricks), and the rest of the wall. Other than that, it is purely decorative.
The picture provided is a bad example but the original purpose of building a thick layer of clay or stone possibly with tiles around it, is to hold the heat from the fire for a long time and provide heat in the room even after the fire itself has finished.
The kind of heat that the stone gives is radiational heat, which is more comfortable to people than actual hot air. (just like radiators do as opposed to convectors) Like an infrared sauna, this makes your body warm even while the air has a lower temperature.
The most efficient and comfortable kind of fireplaces are not open like in the picture, but have a small opening with a door. this creates a more efficient fire, more heat, and more energy stored in the mantel. Albeit less esthetically pleasing perhaps.
The fireplace mantel in an open fireplace has another function of making it possible to heat pots over the fire. If you have a hook under the mantel you can hang pots there to cook food or make tea, etc.
If you are thinking about a way of actually heating your house, you should not choose an open fireplace since the efficiency is only 10% (so you are mostly just heating the air outside your house) as opposed to a closed thick fireplace where the efficiency is up to 80% or 90%