You can look at it in several ways. One is to look at the electric field between the plates (I assume parallel plates). Since there is a dielectric between the plates, the electric field gets reduced and you will need more charge on the plates to increase the voltage between them. Using the definition of capacitance you can figure it out from there.
Dielectrics are materials that are readily polarized. That is, the + and - can be separated a little bit. Sticking a dielectric inside is like sticking another capacitor between your existing widely spaced capacitor.