Redshift is said to be experienced when the light or any other electromagnetic radiations from an object undergo an increase in the wavelength but it does not mean that the light literally becomes red.
Redshift occurs due to the expansion of space. It means the farther away the galaxy is from us, the more space it has expanded in the time.
Because as the light left the galaxy, more of the light has been stretched and so more redshifted the light is.