The meaning of the original statement is that if you have a cubic meter of water (an amount of water that is one meter in height, one meter in width, and one meter in depth), that amount of water would weigh 1000 kilograms.
The way to convert kilograms per cubic meter to grams per cubic centimeter is by using the following formula:
1 kilogram per cubic meter = .001 grams per cubic centimeter
So to find the actual answer, you could set up the equation
1000 = x/.001
So then you multiply both sides by .001
This gets you x = 1
So water has the density of 1g/cm^3