The universe is expanding all the time, so our galaxy has moved relative to other galaxies since the time of the dinosaurs. We can measure the rate at which other galaxies are moving away from us by looking at the light coming from them. Light emitted by objects moving away at high speeds is reddened due to the Doppler effect (if you want more detail on the Doppler effect please ask a follow up question by commenting below).
Almost every other galaxy in the universe is reddened, so we know that they are all moving away from us. The furthest ones are moving fastest, which tells us that the entire universe is expanding. The exception is our nearest neighbour Andromeda, which is moving towards us (so looks bluer) and will probably eventually collide with the Milky Way.
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