Question

How do we know how fast the universe is expanding?

Answer 1

We observe an interesting effect called Doppler effect.

Explanation:

This is actually a very interesting question.
Imagine to have a blinking led at fixed frequency of one blink per second. If you and your led are at a fixed distance you can observe your led and see that every second it turn on and off.

Now the person holding the led start to move away from you. What do you see? You just saw the last blink, you wait for one second and you expect to see the next one. But in the meanwhile your led moved from you, so the led was switched on after 1 second as usual, but the light requires more time to reach you compared to the previous blink and you will see a delay in the following blink.

The delay depends only by the velocity of your led with respect to you, so you have a way to measure the velocity of you know precisely the delay. If the velocity is constant the delay (with respect to your reference that is 1 second blinking) will be constant, if the delay increases it means that your source is accelerating.

This effect, called Doppler effect, is valid for every system that repeats in time. For example, when an ambulance pass out of your windows you can ear the change in the pitch (it becomes higher when the ambulance approaches you and lower when it goes away).
The reason is that the siren is a repeating sound (like the our blinking led) and the source is moving reducing or increasing the delay between two following sounds.

The same thing is done observing the stars. There the thing that repeats in time is the color of the light emitted by a star or a galaxy. The color is like the siren, it is an oscillating wave (light wave actually) that reach us. The frequency of this wave (so the time of repetition) should be fixed and well known, but we see that it changes towards a "lower" color. This tell us that the source is going away from us (like when we listen the lower pitch of the ambulance) and the difference between the observed color and the expected color tell us at which speed the galaxy is drifting away.

This effect is called redshift because the colors move in the direction of the red that has a lower frequency.