Sigh. Is this really so difficult. So let's go step by step. If you have only electromagnetism the world is P, T, C, CP symmstric. The only way some object can have an electric dipole moment then is that it is composed in the usual way by charge distributions. E.g., a water molecule has a large electric dipole moment. Such dipole moments are given as
$$\vec{d}=\int_{V} \mathrm{d}^3 r \vec{r} \rho(\vec{r}),$$
where ##\rho## is the charge distribution.
Now let's consider the time-reversal transformation. By definition the space-time variables transform as
$$t \rightarrow -t, \quad \vec{r} \rightarrow \vec{r}.$$
Electric charge
$$q \rightarrow q.$$
With the transformation properties of ##q## and ##\vec{r}## thus
$$\rho \rightarrow \rho.$$
With the definition of the dipole moment given above this implies
$$\vec{d} \rightarrow \vec{d}.$$
It's T-even, i.e., it doesn't change under T.
Now consider an elementary particle. This can have an electric dipole moment only if this dipole moment is ##\propto \vec{s}##, where ##\vec{s}## is the spin, i.e., an angular momentum. Angular momentum is ##\vec{r} \times \vec{p}##. From the ##T##-transformation properties (we need in addition that by definition ##m \rightarrow m##) it's clar that ##\vec{s} \rightarrow -\vec{s}##, and thus there must be a T-odd contribution to the Hamiltonian ##\propto \vec{E} \cdot \vec{s}##. Now again using the properties of ##T## and the definition of the electric field you get that ##\vec{E} \rightarrow \vec{E}## under ##T## and thus this contribution to the Hamiltonian is ##T## odd and thus violates ##T## symmetry. So you can have an electric dipole moment for an elementary particle only if ##T## (and thus CP) is broken.
Since ##T## and ##CP## is broken by the weak interaction, the electron has, within the standard model, a tiny electric dipole moment due to the weak interaction. This CP violation is too small to explain the matter-antimatter imbalance and that's why one tries to measure the electron's electric dipole moment. So far one is far in accuracy from the predicted dipole moment from the standard model. So the hope is that one finds nevertheless an electric dipole moment which with the given sensitivity then was some orders of magnitude larger than the standard-model value, and this would indicate new physics leading to larger CP violation which then in turn may explain the matter-antimatter imbalance. So far, however, no electric dipole moment of an electron has been found given the sensitivity of the experiments. There's only an upper bound, which is already very remarkable given the difficulty of this measurement.
If you need an RMP to be convinced, take that one:
https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.63.313