“\(I\)” refers to “\(I(\vec r, v)\)” and “\(I'\)” refers to “\(I(\vec r+\delta \vec r, v+\delta v)\)”.
Here, “\(I\)” refers to the intensity of light, “\(\vec r\)” refers to the position of the source, “\(v\)” refers to the radial velocity at which the source moves towards or away from us.
Note that we treat the mixed terms as being very close to zero. This is because we expect the intensity from the background noise to change very little based on position and velocity– that is, \((I_{noise} - I'_{noise}) \approx 0\).