It did not prove this, but it strongly suggested it. Here is how the logic goes: Maxwell's equations for electricty and magnetism predicted wave solutions that travel at a speed of \frac{1} {\sqrt{\epsilon_0\mu_0}}. People plugged in the known values of \epsilon_0 and \mu_0, and got a speed that was exactly equal to the known speed of light within experimental error. This lead to the hypothesis that light was a Maxwell wave. Experiments by Heidrich Hertz showed that Maxwell waves behave like light in every way. This essentially proved that light was a Maxwell wave as much as anything can be proven in physics. Now, the disturbing thing was that you could calculate the speed of a Maxwell wave in one reference frame, and then try to calculate it again, this time in a different reference frame moving with some speed with respect to the first one. The weird thing is that observers in both frames get the same answer. The assumption that was then made by physicists at the time to resolve this apparent paradox was that Maxwell's equations must be stricly valid in only one reference frame. This would mean that only one observer would be correct in applying Maxwell's equations for the speed of the wave and resolve the apparent paradox. This frame was known as the frame of the "luminiferous aether". It was believed that the aether was a fluid that EM waves were carried by.
The natural question that emerged was "what is the aether frame". That is, is the aether stationary with respect to the sun, or some other star, or none of these.(The question was stated this way because, at the time the aether was believed to be a well established fact and a cornerstone of physics) Several experiments were made in an attempt to answer this question, which all failed for one reason or another. Finally, Michaelson developed an experiment which could not fail. He tried to measure the speed of the aether with respect to the Earth by exploting the very small wavelength of light and the resultant fact that a small disturbance could lead to visible results in an interferance pattern.
The experimental set up is much better described by a picture than words, so look at this
http://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/mmexpt6.htm
No matter how the interferometer was rotated, no interferance was observed. It did not matter what season the experiment was performed in, so it couldn't be that the Earth just happened to be moving at the same speed as the aether at that particular moment. It was also disproven that the aether could be dragged around by the Earth based on observations of starlight. The experiment was repeated in labs around the world with the same result: no aether effect. This did not kill the aether, though. The prominent physicist Hendric Lorenz believed that the aether caused one of the arms of the interferometer to contract. In quantifying this contraction, he formulated the relations between space and time in different reference frames that Einstein derived in Special relativity, although Lorenz did not assume the constancy of the speed of light.(These equations are still known as the "Lorenz transformations") However, assuming that the aether does not exist and that the speed of light is the same to all observers results in a simpler theory which ultimately agreed with experiment. If you want me to elaborate on the experiment itself, let me know, but this post is already exteamly long.
