Light, like sound, is a wave. Sound waves are caused by vibrations in the air (or another medium) transmitting energy: if you sit at a single point in the path of a wave and measure the pressure of the air, you’ll see it fluctuate up and down as the wave passes. With light the medium is the strength of the electromagnetic field instead of air pressure, but the principle is the same.

A wave progressing: the points at which the blue dot is at its highest are a set of wavefronts. The height of the dot could represent local air pressure for a sound wave or local EM field for light.
You can think of a wave as being made of a series of ‘wavefronts’ which are emitted from a source and move along in the same direction. For example, all the points at which the pressure is at a maximum make a set of wavefronts for a sound wave. For a simple wave they all move along in a row, all separated by the same, constant distance. This distance is the wavelength of the wave.
Now imagine that the source of the wave is moving away from you. Between emitting one wavefront and the next it will have moved a little distance, and the second wavefront has slightly further to travel. That means it reaches you later than it would have done otherwise, and the wave gets spread out: its wavelength increases. Move the source towards you instead and the opposite happens: the wavefronts get ’squashed’ together and you see a shorter wavelength.
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