Outer space appears dark at night for multiple reasons, the most simple being that the nearest light source, the sun, is blocked by the earth, creating darkness. Outside Earth's atmosphere, outer space appears dark instead of blue because the sun's light is not refracted by atmospheric gases.
The light created by all the stars in the universe is not enough to illuminate space. Stars are far enough apart from each other that the space between them appears dark. The universe is constantly expanding, meaning that every object in space is increasingly farther away from every other object. Because light wavelengths become longer when moving away from something, these light waves appear red from Earth or do not enter the visible spectrum if they are far enough away and are not visible at all.
Until the 20th century, people believed that the number of stars in the universe was infinite, and therefore their light should light up the entire sky, leaving no darkness. The paradox of why this is not the case is called Olbers' Paradox. Because it is now known that the number of stars in the universe is not infinite, the reason for outer space appearing dark can now be explained.