I think a lot of confusion is caused by using the word "ground" to describe the 0volt line in a circuit.
Sometimes the 0volt line is connected to ground, or earth as we say in the UK (and maybe other places as well).
In relation to an antenna, first consider a dipole.
If we connect a transmitter to a feeder cable (co-ax or twin feed) and connect the feeder to the dipole, RF current will flow in the feeder and in each leg of the dipole. The currents in each leg of the dipole will be equal and opposite.
If we disconnect one leg of the dipole, the current in the remaining leg will drop (theoretically) to zero, and the radiated signal strength will also drop to zero.
The current in the leg of the dipole has to flow somewhere, consider it as flowing through space to the other leg of the dipole so that a complete circuit is created.
If we have a situation where it is physically inconvenient to use a dipole, then a single quarter wavelength wire can be used, but the current in that wire has to flow somewhere, so we can use a "ground plane". The ground plane may be the body/chassis/circuit board of the transmitter, or for more predictable results, a sheet of conductive material under the single wire antenna, or often several lengths of wire a quarter wavelength long connected to the screen of the co-ax cable connecting to the antenna.
That is the simple explanation, there are many ifs, buts and complications to what I have just written, this is just the Janet and John version.
While I have been typing it also occurs to me that the expression "groundplane" is also used to describe a continuous layer of copper on a circuit board. Usually where high frequency signals are involved.
The groundplane is usually used as the 0volt return for the DC and signals, and because it is one sheet of copper, parasitic inductance low preventing unwanted coupling between various parts of the circuit. (Which is why high frequency circuits dont work on breadboards).
JimB