How do we know how fast the universe is expanding?

This is a really fascinating question. Let's say you have a blinking led that blinks once every second at a fixed frequency. If you are standing a certain distance away from the led, you can watch it turn on and off once every second.

Now that the person holding the led is starting to move away from you, what do you see? After the last blink, you wait a second expecting to see the next one, but in the interim, your led moved away from you, so the led turned on as usual after a second, but it takes longer for the light to get to you than it did in the blink before, so the blink after that will be delayed.

The only factor that determines the delay is the velocity of your led relative to you, so you can measure the velocity to determine the exact delay. If the velocity remains constant, the delay (measured in terms of one second of blinking, your reference) will also remain constant; if the delay increases, it indicates that your source is accelerating.

This effect, known as the Doppler effect, is applicable to any system that repeats in time. For instance, you can hear the pitch change when an ambulance passes outside your window—it gets higher when it gets closer and lower when it moves away. This is because the siren is a repeating sound, much like our blinking led, and the source is moving, which either shortens or lengthens the time between two subsequent sounds.

Similar observations are made of stars, where the time-varying element is the color of light emitted by a star or galaxy; this color is analogous to a siren, an oscillating wave (actually a light wave) that reaches us; its frequency (and thus its repetition time) should be fixed and well-known, but we observe that it shifts towards a "lower" color, which indicates that the source is moving away from us (much like when we hear the ambulance's lower pitch). The discrepancy between the observed and expected colors indicates the speed at which the galaxy is moving away.

Because the colors shift in the direction of the red, which has a lower frequency, this effect is known as redshift.

A number of techniques are used to measure the cosmic microwave background radiation, observe the redshift of distant galaxies, and analyze the brightness of Type Ia supernovae in order to determine the rate of expansion of the universe.