This is called Olber's Paradox. The idea is that in an infinite universe where galaxies are uniformly distributed in space, the diminishment of the received light by the inverse square law is compensated by the increase in the number of galaxies at each surface area element, so the brightness of the sky should increase until it has the same emissivity as the surface of star.
But, the main reason we don't see this is because the universe is simply not old enough for us to have received the light from the most distant and numerous contributors to this seemingly divergent 'integral'. Also, the cosmological redshift shifts the received light from the optical band into the infrared and microwave band, but this isn't fair because although it causes distant galaxies to be come invisible to the eye, the light energy from them still adds up to a divergent sum.