Question

When blue shift occurs, how does the wavelength of light that we see change?

Answer 1

The doppler effect changes the wavelength of the light emitted, depending upon whether source is moving away or coming towards the detector.
Doppler effect in light is actually a relativistic effect (caused due to relativistic time dilaton and can be explained by special theory of relativity) but somewhat similar to the one which happens in sound waves. This effect has nothing to do with the bending in spacetime.
When the source is moving away from the detector, the wavelength of the light emitted from the source appears to be increased as seen by the detector (think of a wave being stretched), as a result the frequency decreases (since, speed of light is constant in all frames). Due to this decrease in frequency the light emitted from the source appears more red (since red color is on low frequency side in the electromagnetic spectrum) compared to the situation when source was at rest.
Vice versa for the source moving towards the detector
This shift in frequency is also known as doppler shift.
The shift in frequency when the source is moving away is known as redshift and the opposite one is known as blueshift.