Assuming it was mounted appropriately in the kettle, and assuming that you changed the connector from the NEMA L6-30 to the NEMA 5-15 as used for 120V outlets in the US, yes, you could simply plug it directly into the outlet, and it would run at 100% power. Note that if it's a 240V heating element rated for X Watts, you're only going to get X/4 watts out of it at 120V.
There is no "building" of the heating element; it's literally just a piece of metal with places to connect wires to, a big resistor. But it does need to be mounted correctly to prevent liquid from getting to those connections.
For the amount of power you can get out of residential 120V, you probably do want to run the element at "full power" for the whole boil, so you can get away with just plugging it into the wall. But, the electric brewery setup does assume that you're going to run the boil-kettle element at less than full-time. You don't need as much energy to maintain a boil as to raise to boiling temps. They accomplish this, of course, with a PID controller and SSR to switch the element on and off periodically.
If you just want something you can throw into the kettle, look into "bucket heaters", which are basically the same, but designed A/ to run on 120V and B/ to be submerged in liquid. You could even use 2 at a time, if you can find multiple outlets close enough to the kettle.