Current and voltage are not especially good to present what happens in an antenna. The actual thing that happens in an antenna which is connected to a transmitter is that a radiowave comes along the feeding cable (it's in the space between the wires, in coaxial cable it's between the shield and the center wire). In antenna the fields spread to wider area and if the antenna is properly designed a substantial part continues to the free space around the antenna as wanted. The rest is dissipated in material losses or reflected back towards the transmitter.
The key thing is that there's a propagating wave around the antenna and some reflection happens at the far end from the feeding point. The current which the fields of the wave induce to the metal cannot jump out of metal in usual power levels, so the end of the metal surface is a discontinuity which causes reflection.
From the theory of transmission lines we should remember that when a wave meets a strict "no current possible any more" border the electric fields of the arriving and reflecting waves are in every moment to the same direction at that border, so the sum field of the arriving and reflecting waves is as strong as possible with that power level.
With transmission lines we usually call the RMS value of the sum field "standing wave" It's maximum is at the unconnected end of the line.
On the surface of an antenna less far from the feeding point the fields of the different direction propagating waves partially cancel each other (but do not disturb each other by any means) so the sum field can be weaker.
At the same time when waves propagate along the surface of the antenna they also lose energy to the space as radiated wave. That radiation is attempted to be maximized in antenna designs, in transmission lines it's wanted to be kept zero.
Voltage is a coarse measure for electric field, all vector field structure is omitted, but the proportionality to electric field strength is still there. Thus also voltage between the ends of a dipole antenna can be higher than at the feeding point.