The Doppler effect describes shifting of signals by movement of the source towards or away from an observer. Light is shifted to the red end of the spectrum when objects emitting it are moving away from us and to the blue end of the spectrum when objects are moving towards us. Which means shifting to the red end indicates movement away from us and to the blue end movement towards us.
If we were living in a static universe with trillions of stars we should either observe no shifting at all, which means no movement, or equally distributed shifting to both ends if stars simply were moving in more or less random directions. What we however find is the vast majority of stars show a red shift which means that the vast majority of stars are moving away from us. One way to explain that would be expansion of space which would be support for the Big Bang theory.