How does the sun illuminate the space station when it is on the opposite side of the earth at night?

It doesn't. The space station falls into darkness. However, you can still see it for a while if you catch it passing overhead just after the sun goes down or just before sunrise because, being so high above the Earth, it catches the sunlight for longer than someone on the ground and picks it up in the morning before anyone else. Neverthe less, you will never see the station passing over at midnight, because just like you and me, it's in the shadow of the Earth.