Why is scale space (DoG) needed to detect scale invariant features?
As you know, you find interest point (SIFT points) by finding the local maxima in scale-space, it mean the response of the detector must be maximum regarding the coordinates and also the scale. So knowing the scale you select features from that scale. Now consider the same process for same interest point in the same picture but with different zoom, the detector maxima will be in the same scale, so it will collect the same feature. So in the matching these two same feature will be matched together.

So the scale invariant is a result of finding maxima with respect to scale.