Let's say you have an insulator that is electrically neutral(has no net charge). Let's say you are able to add additional electrons into the same insulator resulting in the insulator having a net negative charge. These electrons that were added to the insulator will stay where they are and not move. How is this possible when same charges repel each other? Shouldn't those excess electrons added to the insulator repel away from each other?
Consider the following diagram of the quantum mechanical densities of electron states in different types of material:
The Fermi energy $E_F$ is the energy level new electrons are added at. Loosely said, the states below this energy are filled, the states above are empty, with a smooth transition around $E_F$. The conducting electrons are those in this smooth transition.
On the far left is a metal. There are plenty of states available around $E_F$, so adding an electron simply puts it among the already conducting electrons.
On the far right is an insulator. The Fermi energy is in a gap between the bands with no available electron states. The lower band is filled and cannot conduct, the upper band is empty and cannot conduct. If you add an electron to this material, you can only add it in the upper band, since the lower band is already full. When you do this, you effectively lift $E_F$ up to the upper band, and you no longer have a perfect insulator. The electron you added can "move" just fine, and conduct electricity. This is similar to the n-type situation in the diagram.
It is possible to apply a force without having any motion (that should be obvious just from your day to day experiences, right?). So there is a repulsive force on the electrons but they don't move anywhere, at least nowhere fast because the force is not enough to push them through the insulator...at least until the insulator breaksdown.
Like sticking a bunch of people who hate each other into the same room full of stuff. They might want to get as far from each other as possible, but they can't go anywhere.
If one charges an insulator, then yes, sometimes the excess charge stays on the insulator. A charged particle is attracted to any polarizable medium (static cling, in your socks, is this effect, writ large). Mobility of charge in such a situation is dependent on the LOCAL electric field, not the large-scale "it has multiple positive charges" global situation. The polarized medium puts a local attractive charge next to the excess charge, after all, at the expense of slightly altering (polarizing) the bound-to-molecules charges in their molecular orbits.
The applied charges do NOT "stay where they are" however, they migrate to the most attractive local sites (and will slowly drain away into air or elsewhere, as thermal motion jostles them about). This effect, leakage, is slow, but important. Detectors of ionizing radiation are basically looking at the excess leakage that happens due to photons or other energetic particles complicating the insulator's inherent insulation property.