How does red shift provide information about the speed of galaxy?

Let's begin by analogy with sound waves. You may have stood by a railroad crossing while an approaching train was sounding its horn. As the horn passes you, you'll note that the pitch of the sound (= frequency of the sound wave) changes to a lower tone. What's happening is that, as the horn approaches you, it moves a bit closer to you as it emits each peak of the sound wave's frequency, thus crowding the peaks of the waves closer together and increasing the frequency, which your ear hears as a higher pitch. As the horn moves away from you, the opposite happens -- the horn moves a bit farther away as it emits each peak of the sound wave, thus stretching out the frequency and presenting a lower pitch to your ear.

Light waves do the same thing; it's just that the speeds required to make a significant change in the frequency of the light wave (= the color of the light) are much higher than with sound. As an object emitting light approaches, the frequency of the light is higher than it would be if the object was at rest relative to you -- and the frequency of the light is lower than the rest frequency if the object is moving away.

Since blue light has a higher frequency than red light, frequency increase is referred to as "blue shift" and frequency decrease is called "red shift", regardless of the actual colors of light involved. So even radio-frequency waves can be "redshifted".

How can this be measured? It turns out that each chemical element emits its own pattern of very specific, narrowly-defined frequencies of light when heated to incandescence. We can measure those frequencies in a laboratory, and then compare them to the frequencies that we actually measure from those same chemical elements in the light coming from a galaxy (hydrogen, for example, has a strong and easily-recognized pattern of frequencies).

Knowing what the frequency was in the laboratory, and measuring the frequency as it comes from a galaxy, gives the numbers we need to plug into a formula to calculate how fast the galaxy is moving along the line of sight from us to the galaxy. If the galaxy light's frequency is higher (blue shift), the galaxy is moving toward us -- the amount of the difference between the laboratory light and the galaxy light frequency tells us exactly how fast. Similarly, redshifted light from the galaxy (by far the more usual case) tells us how fast the galaxy is moving away from us.