The purpose of the MM experiment was to measure he velocity of the earth through the aether, which was the proposed medium for light waves to propagate through. The aether was devised because physicists postulated that since light is a wave, it needed a medium to travel through, e.g. water for water waves, air for sound waves.
MM devised an experiment to measure the velocity of the earth through the aether, but they did not seek to prove or disprove it's existence. They were firm believers in the aether until their experiment yielded a null result. Actually, I am pretty sure it took even more convincing.
The experiment involved having two light beams fired at right angles to each other while colliding with a half silvered mirror which allowed half of the light to pass through, and the other half to be deflected. Since the beams were fired at right angles to one another, it was thought that one beam would be traveling with the aether and one would be travelling across it. Think about a river. If you a swimming against the current, and then with the current, it is hard initially, but is very easy on the way back because you would pick up the velocity of the water, while swimming across the river would be equally hard both forward and back. This idea was the premise for the MM experiment.
After they conducted the experiment the first time, they did a second time after rotating the apparatus through 90 degrees ideally switching the two beams positions. Because the beams of light cannot be perfectly placed an equal distance from the half silvered mirror, they expected the interference pattern to change after rotating the apparatus. This did not occur. Absolutely no change in the interference pattern was observed. This provided experimental evidence for the idea that light did not need a medium to travel through.
Hope that helps!