The purpose of dosage compensation is to even out the sex chromosomes. Females have two X's and males have an X and a Y. The Y chromosome is considerably smaller, which means there are much less genes on it. To make everything even, one of the females X chromosomes is condensed. This turns off that X chromosome, and under a microscope it appears as a small black spot in the cell's nucleus (called a Barr Body).
What do you think some of the side effects are for females when one X chromosome is "turned off"? How do you think scientists were able to prove this?