The term "scale-invariant" means the following here.
Let's say you have image I, and you have detected a feature (aka an interest point) f at some location (x,y) and at some scale level s. Now let's say you have an image I', which is a scaled version of I (downsampled, for instance). Then, if your feature detector is scale-invariant, you should be able to detect the corresponding feature f' in I' at the corresponding location (x',y') and corresponding scale s', where (x, y, s) and (x', y', s') are related by the appropriate scaling transformation.
In other words, if your scale-invariant detector has detected a feature point corresponding to someone's face, and then you zoom in or out with your camera on the same scene, you should still detect a feature point on that face.
Of course, you would also want a "feature descriptor" which would allow you to match the two features, which is exactly what SIFT gives you.
So, at the risk of confusing you further, there are two things that are scale-invariant here. One is the DoG interest point detector, which is scale-invariant, because it detects a particular type of image features (blobs) irrespective of their scale. In other words, the DoG detector detects blobs of any size. The other scale-invariant thing is the feature descriptor, which is a histogram of gradient orientation, which stays more or less similar for the same image feature despite a change in scale.
By the way, the difference of Gaussians is used here as an approximation to the Laplacian-of-Gaussians filter.