Look up the Michelson-Morley experiment.... Basically Michelson, a Russian immigrant, Annapolis graduate and naval officer was recognized as the best experimental optical physicist in the world at the time. He invented the Michelson interferometer to measure any change in the speed of light from the Sun in the morning when the Earth's rotation was towards the Sun and in the evening, when the Earth was rotating away from the Sun. Given the sensitivity of his apparatus this should have been a piece of cake. (The experiment is not unprecedented, Bradley in 1727 had observed that the apparent position of a star varies with the direction of the Earth around the Sun, much as the apparent direction of rain is changed when we drive through the rain in a car.)
He was surprised to discover that there was absolutely no difference. No theory could adequately account for this result until Einstein's theory of relativity. Einstein held that the speed of light - derived from Maxwell's Equations and predictable from measurements of the magnetic and electical properties of space that apparently had nothing to do with the speed of light - were fundamental physical quantities and the same in any non-accelerated reference frame.
One prediction of the theory of relativity is that if you were stationary with respect to another observer and he and you each established a laboratory and measured the speed of light you would agree. If he and his laboratory was later moving at, say, half the speed of light with respect to you and you shone a laser beam in his direction, both of you would measure the same speed for the mutally observed laser light as you did before, even though you both would agree that each of you was moving with respect to the other.
I know it sounds counter-intuitive, but it turns out that's the still the best explanation of things that are observed every day. You either have to take it on faith or learn more physics.