Your question, which seems simple, is actually very difficult to answer! It is a question that many scientists pondered for many centuries - including Johannes Kepler, Edmond Halley, and German physician-astronomer Wilhelm Olbers.
There are two things to think about here. Let's take the easy one first and ask "why is the daytime sky blue here on Earth?" That is a question we can answer. The daytime sky is blue because light from the nearby Sun hits molecules in the Earth's atmosphere and scatters off in all directions. The blue color of the sky is a result of this scattering process. At night, when that part of Earth is facing away from the Sun, space looks black because there is no nearby bright source of light, like the Sun, to be scattered. If you were on the Moon, which has no atmosphere, the sky would be black both night and day. You can see this in photographs taken during the Apollo Moon landings.
So, now on to the harder part - if the universe is full of stars, why doesn't the light from all of them add up to make the whole sky bright all the time? It turns out that if the universe was infinitely large and infinitely old, then we would expect the night sky to be bright from the light of all those stars. Every direction you looked in space you would be looking at a star. Yet we know from experience that space is black! This paradox is known as Olbers' Paradox. It is a paradox because of the apparent contradiction between our expectation that the night sky is bright and our experience that it is black.
Many different explanations have been put forward to resolve Olbers' Paradox. The best solution at present is that the universe is not infinitely old; it is somewhere around 15 billion years old. That means we can only see objects as far away as the distance light can travel in 15 billion years. The light from stars farther away than that has not yet had time to reach us and so can't contribute to making the sky bright.
Another reason that the sky may not be bright with the visible light of all the stars is because when a source of light is moving away from you, the wavelength of that light is made longer (which for light means more red.) This means that the light from stars that are moving away from us will become shifted towards red, and may shift so far that it is no longer visible at all. (Note: You hear the same effect when an ambulance passes you, and the pitch of the siren gets lower as the ambulance travels away from you; this effect is called the Doppler Effect).