You usually do that in multilayer designs where you designate specific layers to certain power nets such as 3V3 or GND. the intended goal is to make the supply and return path of the current as short as possible (also, the two layers form some kind of small capacitance). If it's on the outside layers, it's sometimes done to block RF from leaving the inner layers. Also, it's nice for production since it reduces the amount of copper which needs to be etched away.
Well, most of all: it's a combination :-)
In Altium you might not want to use fills but instead you will be using polygons where you can specify the connect style (if you want the connection to be a thermal relief). Also, using the Layer Stack Manager you can specify specific layers to be power planes which basically makes them "inverted". So as long as you do not draw a line on such a layer, the full layer will be connected to a specific net. By drawing lines (thereby separating two areas) you can create so called split-power planes.