Ok back to school here... The reason space does not light up is that it is a void/vaccume with very little in it. Light is only visable when reflects off matter such as the moon. Light travels from the sun (which is a very large ball of mass), this light is pure bright white and can not be seen with the human eye... It travels in straight lines (in frequencys actually but its straight as far as you are concerned) this then hits an object like the moon and the light is bounced back to you who is staring up up at it in the night sky..
The reason the sky lights up is the sky is and atmosphere made up of gasss, particually the o-zone layer, when light filters through it as it comes straight from the sun the light alters and the eye sees this as the bright blue sky on a clear day. When its night time your in the shadow of the half of the planet in the day time so the light does not make its way through the atmosphere and light it up (in basic terms)