How can current flow in dipole antennas when there is no potential across them?
The rule that says a wire has the same potential at every point is a rule for lumped circuits. The lumped circuit approximation applies when the dimensions of the circuit are much smaller than the wavelength associated with the frequencies of the signals present. An antenna, on the other hand, typically must have a dimension that is a substantial fraction of the wavelength, for example 1/4 or 1/2 wavelength. Therefore an antenna is, practically by definition, not a lumped circuit, so that rule doesn't apply. 

In fact there will be a potential difference (disregarding the fact that you have to be careful about even using the concept of potential in a distributed circuit)  between the points along the antenna, which pushes the charge around and produces the radiation.


  the animation shows the ac current when there should not be any since no potential exists.


The red curve in the animation shows how the potential varies along the antenna. It shows there absolutely is a potential difference between different locations along the antenna.

It is true that the potential difference goes to zero at the instant when the currents are at a maximum, and vice versa. This phase difference comes about because the antenna has both inductance and capacitance, and again because the dimensions of the antenna are on the same order of magnitude as the wavelength of the signal.


  I become confused when I consider the scenario where there is a current carrying wire and I stood on a plastic stool and touched the wire does it mean that current will flow to me as well even though I am at the same potential as the wire?


The current flowing to you is why you reach the same potential as the wire. Before you touched the wire you were at some other potential. When you touched it, some charge flows in to (or out of) your body to the wire, and that is why you achieve the same potential as the wire.