The Michelson-Morley (MM) experiment is influential because it falsified the aether theories of the 19th century and thus lead to the development of the Special Theory of Relativity (STR).
Up until the 1970s, the common view has been that the MM experiment provided the measurements on which STR is based. And if STR is based on these measurements, then they can also serve as evidence for the theory.
The null result of the experiment has two important implications:
- First, it implies that the earth is not moving through space, as only observers at rest with respect to the vacuum of space should measure an isotropic speed of light.
- Next, provided that the earth is in fact moving through space, the null result implies that the speed of light is independent of the motion of the observer, as apparently even observers in motion with respect to the vacuum of space measure an isotropic speed of light.
The first of these implications has been interpreted to mean that the motion of the earth through space is undetectable - and more generally, that observers cannot detect their own motion. Hence, motion in STR is strictly relative, meaning that you always need a reference point for motion; without a reference point, you can't tell whether you are moving or not (as suggested by the null result of the MM experiment).
The second implication is the basis for the "principle of the constancy of the velocity of light", as defined by STR.