The voltage across the capacitor has to stay the same since it is connected to a fixed voltage supply, which means that the potential before insertion and after insertion is equal. That would mean that the electric field within the capacitor is also equal before and after (since E = -dV/dR).
However, when a dielectric is inserted, it reduces the field since the molecules of the dielectric align themselves in such a way that the moment is opposite to the external electric field, which is also supported by:
K = Eexternal/Ereduced
where K is the dielectric constant.
So when it is connected to the battery, why does the electric field remain the same? What happens within the capacitor and dielectric?