Laboratory experiments here on earth have determined that each element in the periodic table emits photons only at certain wavelengths. these photons are apparent to astronomers as either emission or absorption lines in the spectrum of an astronomical object, like stars in galaxies. astronomers can, by measuring the position of these spectral lines, determine which elements are present in the star itself or along the line of sight. when astronomers perform this analysis, they note that for most astronomical bodies, the observed spectral lines are all shifted to longer (redder) wavelengths. this is known as cosmological redshift and is analogous to the doppler redshift. what does this redshift indicate to us about the universe in general?

The only thing we know about so far that can shift wavelengths of light
to longer wavelengths is when the source of the light is moving away
from the observer.

When we look at the light from distant galaxies, the light from them is
always shifted to longer wavelengths than it SHOULD have. 

AND ... The farther away from us a galaxy IS, the MORE its light is
shifted to wavelengths longer than it should have.

So far, this indicates to us that the whole universe is expanding.
That's the only way to understand what we see, because that's
the only thing we know of that can shift light to longer wavelengths.

By the way ... The most interesting thing about these observations
and measurements is:  When astronomers see this light from distant
galaxies and measure the wavelengths, how do they know how far
the wavelengths shifted ?  How do they know what the wavelengths
SHOULD be ?
 
I'll leave you to read about that in the next few years.