How do we know the universe is expanding, and what does that actually mean for us?
We know because of Hubbles Law (discovered in 1929): almost every galaxy is moving away from us, and the farther away it is, the faster it recedes. This is measured by cosmological redshift the light from distant galaxies is stretched to longer (redder) wavelengths as space itself expands while the light travels. The simple formula is v = H d (velocity = Hubble constant distance).
Its not that galaxies are flying through space like bullets the fabric of space between them is stretching, like dots on an inflating balloon. There is no center; every galaxy sees the same thing.
For us: The expansion is accelerating (thanks to dark energy). In the far future, distant galaxies will recede beyond our observable horizon and disappear. The universe will slowly become colder, darker, and more isolated the Big Freeze. Local groups like ours stay bound by gravity, though.

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