The current accepted answer is that, although the universe may be infinitely large, it is not infinitely old. It is thought to be about 13.8 billion years old, so we can only see objects as far away as the distance light can travel in 13.8 billion years. Light from stars farther away has not reached Earth, and cannot contribute to making the sky bright. Furthermore, as the universe is expanding, many stars are moving away from Earth. As they move, the wavelength of their light becomes longer, through the Doppler effect, and shifts toward red, or even becomes invisible. As a result of these two phenomena, there is not enough starlight to make space anything but black.