SIFT: why it is invariant to affine tranlsation?
I consider that you already know the proof of how we get DoG from LoG, so I won't go further with it. 

According to your question "how by approximation,using DoG, we get rid of that scale factor in SIFT". DoG works as a feature detector to detect blobs in the given image (blob like shapes) which you can say it is a bit kind of edge detectors here is a photo of some blobs detected using DoG:



These blobs detected at different scales using the scale space pyramid. Using larger values of sigma you get bigger blobs. Experimentally DoG give you the best notion of scale as you can see here :


So, experimentally DoG gives you the most robust features that are scale invariant by constructing the scale space and computing the DoG and finally extracting local maxima points.

Useful links: 

1- Lecture PPT

2- Best known video lecture for SIFT

NOTE: This not a professional answer and I hope to hear more from other fellows