Your mouth, throat and nasal cavities are are part of an "air chamber" that your voice resonates in. Since your voice is made by air moving past your vocal chords - that air has to go out someplace. Usually your mouth and nose. That flow, shape, and volume of those spaces all determine how the sound made by your vocal chords is 'shaped'.
When you plug your nose - you change the air flow and the resonance of that cavity - you make it different. Hence your voice sounds different.
You can experiment by using a vacume hose or paper towel tube. By holding one against your mouth or nose and speaking through it - you've again changed the size and shape of that cavity - by adding the tube. Again, it'll make your voice different.