Fill the larger pot about half full with soil, wet it a little and mix it around. To transplant, just hold your hand over the cup with the main stem in between your fingers then flip it upside down, the whole thing should slip right out of the cup. You can put it directly in the larger pot or you can break up the roots a little with your hands. Then add more soil around the sides/top to fill in the space.
Edit: make sure the soil is wet, but not muddy